Probability of Independent and Dependent Events (6.2)
Summary
TLDRThis video script explores the concepts of independent and dependent events in probability. Independent events, such as rolling a die and flipping a coin, have outcomes that don't affect each other's probabilities. The script explains how to calculate the probability of both occurring with a simple formula. Dependent events, on the other hand, like drawing marbles without replacement, change the probability of subsequent events. The video provides examples and calculations for both scenarios, emphasizing the importance of understanding event dependence in probability.
Takeaways
- đČ Independent events are those where the occurrence of one event does not influence the probability of another event. For example, rolling a die and flipping a coin are independent because the result of one does not affect the other.
- đ The formula for calculating the probability of two independent events occurring together is the product of their individual probabilities: P(A and B) = P(A) * P(B).
- 𧩠To determine the probability of an event, you divide the number of favorable outcomes by the total number of possible outcomes.
- đą For the example given, the probability of rolling a 5 on a 6-sided die is 1/6, and the probability of getting heads on a coin flip is 1/2.
- đ€ Dependent events are the opposite of independent events; the occurrence of one event affects the probability of another. An example is drawing marbles from a box without replacement.
- đ The probability of drawing a green marble first and then a blue marble from a box of 10 marbles (7 green, 3 blue) changes after the first draw due to the reduction in total marbles and the specific color count.
- đ The probability of drawing a blue marble after drawing a green one without replacement changes from 3/10 to 3/9, reflecting the dependency of the events.
- đ To calculate the probability of two dependent events, you multiply the probability of the first event by the probability of the second event after the first has occurred.
- đ The probability of drawing two green marbles without replacement is calculated by considering the change in the total number of marbles and the number of green marbles after the first draw.
- đ Understanding the difference between independent and dependent events is crucial for accurately calculating probabilities in various scenarios.
- đ The video provides a clear explanation of how to calculate probabilities for both independent and dependent events, using examples to illustrate the concepts.
Q & A
What are independent events?
-Independent events are those where the occurrence of one event does not affect the probability of another event. For example, rolling a die and flipping a coin are independent events because the outcome of the die roll does not influence the coin flip.
How do you calculate the probability of two independent events happening together?
-To calculate the probability of two independent events happening together, you multiply the probability of the first event by the probability of the second event. The formula is P(A and B) = P(A) * P(B).
What is the probability of rolling a 5 on a die and getting heads on a coin flip, given that both are independent events?
-The probability of rolling a 5 on a 6-sided die is 1/6, and the probability of getting heads on a coin flip is 1/2. To find the combined probability, multiply these two probabilities: (1/6) * (1/2) = 1/12 or 0.0833.
What are dependent events?
-Dependent events are those where the occurrence of one event affects the probability of another event. An example would be drawing two marbles from a box without replacement, where the outcome of the first draw affects the probability of the second.
How does the probability change when drawing two marbles without replacement from a box containing green and blue marbles?
-The probability changes because the total number of marbles and the number of marbles of each color decrease after the first draw. For instance, if you draw a green marble first, the probability of drawing a blue marble second is now 3/9 instead of 3/10.
What is the correct formula to use when calculating the probability of dependent events?
-For dependent events, the correct formula is P(A and B) = P(A) * P(B after A). This means you multiply the probability of the first event by the probability of the second event occurring after the first event has already occurred.
What is the probability of drawing two green marbles without replacement from a box of 10 marbles, with 7 green and 3 blue marbles?
-The probability of drawing the first green marble is 7/10. If the first marble drawn is green, there are now 6 green marbles left out of 9 total marbles. The probability of drawing a second green marble is 6/9. The combined probability is (7/10) * (6/9) = 7/15 or 0.4667.
Why is it incorrect to use the formula for independent events when dealing with dependent events?
-It is incorrect because the formula for independent events assumes that the outcome of one event does not affect the outcome of the other. In dependent events, the outcome of the first event changes the conditions for the second event, thus altering its probability.
What is the difference between drawing with replacement and drawing without replacement?
-Drawing with replacement means that after an item is drawn, it is put back into the set before the next draw, keeping the total number of items constant. Drawing without replacement means the item is not put back, thus reducing the total number of items available for subsequent draws.
How can you determine if events are independent or dependent?
-You can determine if events are independent by checking if the outcome of one event has no effect on the probability of the other event. If the outcome of one event changes the probability of the other, then the events are dependent.
What is the probability of drawing a green marble and then a blue marble without replacement from a box with 10 marbles, 7 green and 3 blue?
-The probability of the first event, drawing a green marble, is 7/10. After drawing a green marble without replacement, the probability of the second event, drawing a blue marble, is 3/9. The combined probability is (7/10) * (3/9) = 7/30 or 0.2333.
What is the significance of the number of favorable outcomes and total possible outcomes in calculating probabilities?
-The probability of an event is calculated by dividing the number of favorable outcomes by the total number of possible outcomes. This gives the likelihood of the event occurring in a single trial.
Outlines
đČ Understanding Independent and Dependent Events
This paragraph introduces the concepts of independent and dependent events in the context of probability. Independent events are defined as occurrences where the result of one does not influence the probability of another. An example given is rolling a die and flipping a coin, which are separate and do not affect each other's outcomes. The formula for calculating the joint probability of two independent events is the product of their individual probabilities. This is demonstrated through an example where a die is rolled and a coin is flipped, and the combined probability of rolling a 5 and getting heads is calculated. Dependent events are then contrasted, where the occurrence of one event alters the probability of another, using the example of drawing marbles from a box without replacement, which affects the probability of subsequent draws.
đ Calculating Probabilities of Dependent Events
The second paragraph delves deeper into dependent events, explaining that they are influenced by the outcomes of previous events, often seen in scenarios without replacement. Using the marble-drawing example, the paragraph clarifies the mistake of applying the independent events formula to dependent events. It then correctly calculates the probability of drawing a green and then a blue marble without replacement, showing how the probability changes after the first draw. Another example is provided, calculating the probability of drawing two green marbles consecutively, emphasizing the need to adjust the formula to account for the altered probabilities after each event. The paragraph concludes with a recap of the differences between independent and dependent events and how their probabilities are calculated, and ends with a call to support the creators on Patreon and visit their website for more resources.
Mindmap
Keywords
đĄIndependent Events
đĄDependent Events
đĄProbability
đĄFavorable Outcomes
đĄTotal Outcomes
đĄWithout Replacement
đĄFormula
đĄDice
đĄCoin Flip
đĄMarbles
đĄPatreon
Highlights
Definition of independent events with respect to probability.
Example of independent events: rolling a die and flipping a coin.
Explanation of how the outcome of one event does not affect the other in independent events.
Formula to calculate the probability of two independent events happening together.
Example calculation of rolling a 5 on a die and getting heads on a coin.
Method to determine the probability of an event using favorable outcomes and possible outcomes.
Calculation of the probability of drawing a green and then a blue marble without replacement.
Clarification on the incorrect use of the independent events formula for dependent events.
Explanation of dependent events and how they differ from independent events.
Impact of drawing without replacement on the probability of subsequent draws.
Calculation of the probability of drawing two green marbles without replacement.
Modification of the formula for dependent events to account for the change in probability after an event occurs.
Recap of the key differences between independent and dependent events.
Support for the video creators through Patreon and website access.
Access to study guides and practice questions on the creators' website.
Transcripts
in this video we'll be talking about
independent events and dependent events
both of these events will be defined
with respect to probability what are
independent events independent events
refer to the occurrence of one event not
affecting the probability of another
event for example let's say we are
rolling a die and flipping a coin both
of these are two separate events we can
say that the first event is rolling or
die and the second event is flipping a
coin because the outcome of the first
event does not affect the outcome of the
second event these events are said to be
independent events in other words
rolling a six doesn't increase or
decrease the probability of a coin
landing on heads or tails the
probability of getting heads is 0.5 and
it stays that way regardless of what you
roll to calculate the probability of two
independent events happening together
you can use this formula where the
probability of a and B is equal to the
probability of event a times the
probability of event B let's do an
example
if you roll a 6-sided die and flip a
coin what is the probability of rolling
a 5 and getting heads the first thing we
should do is write down the formula but
in order to use this formula we need to
know the probabilities of each event if
you watch the previous video you should
know that the probability of an event is
equal to the total number of favorable
outcomes divided by the total number of
possible outcomes for the first event
there is only one favored outcome which
is rolling a 5 and there are our total
of six possible outcomes since we are
rolling a six-sided die as a result the
probability of rolling a 5 is equal to 1
over 6 for the second event we know that
the probability of getting heads is
equal to 1 over 2 or 50% and we know
this because there is only one desired
outcome which is getting heads and there
are a total of two possible outcomes
since the coin can land on either heads
or tails now that we have the
probabilities for each event we can use
the formula and all we have to do is
multiply them together 1 over 6 times 1
over 2 give
an answer of 1 over 12 as a result the
probability of rolling a 5 and getting
heads is equal to 1 over 12 or 0.08 33
what are dependent events dependent
events are simply the opposite of
independent events dependent events
refer to the occurrence of one event
affecting the probability of another
event for example suppose we have a box
that contains 10 marbles 7 other marbles
are green and three of the marbles are
blue
based on this we know that the
probability of drawing one green marble
is 7 over 10 or 0.7 and the probability
of drawing one blue marble is 3 over 10
or 0.3 if we randomly select two marbles
from this box what is the probability of
drawing a green marble and then a blue
marble with our replacement a common
mistake in solving this problem is by
using the formula and then multiplying
the probabilities of each marble
together so you'll have 7 over 10 times
3 over 10 however this process is
incorrect this formula can only be used
for independent events and we know that
this is not an independent event since
the marbles are being drawn without
replacement the term without replacement
means we are drawing the marble without
putting it back into the box this means
that the probability will change after
every draw as a result this is a
dependent event where the probability of
one event affects the probability of
another event in other words drawing the
first marble affects the probability of
the next marble let's see why this is
for the first event there are 10 marbles
in the box and since we have a total of
7 green marbles the probability of
drawing one green marble is 7 over 10 or
0.7 for the second event the probability
of drawing a blue marble is not 3 over
10 since there is a total of nine
marbles left in the box
with a total of 3 blue marbles remaining
the probability of drawing a blue marble
is now equal to 3 over 9 or 0.33 as you
can see the probability of drawing a
blue marble has changed at first it had
a value of 0.3 but now it has a value of
0.33 or 3 over 9 as a result this is a
dependent event because the occurrence
of the first event affected the
probability of the second event now to
finish this problem all we have to do is
multiply these two values together seven
over ten times three over nine gives us
an answer of seven over thirty or zero
point two thirty three let's do another
example using the same scenario what is
the probability of drawing two green
marbles without replacement feel free to
pause the video so you can try this
question for yourself to solve this
question we we use the formula except we
have to make some modifications to it
the probability of a and B is equal to
the probability of a time's the
probability of B after event a has
occurred I will assign event a as
drawing the first screed marble and I
will assign event B as drawing the
second green marble the probability for
drawing the first green marble is equal
to 7 over 10
since the box is untouched if this event
was successful there will be six green
marbles remaining with a total of nine
marbles left in the box therefore the
probability of drawing the second green
marble is equal to 6 over nine and
finally to get the answer all we have to
do is multiply these two values together
7 over 10 times 6 over 9 gives us an
answer of 7 over 15 or 0.46 67 to
quickly recap for independent events the
outcome of one event does not affect the
outcome of the other event if events a
and B are independent the probability of
a and B occurring is equal to the
probability of a time's the probability
of B and for dependent events the
outcome of one event does influence the
outcome of the other event this is
commonly seen when drawing items are not
returned if events a and B are dependent
the probability of a and B occurring is
equal to the probability of a time's the
probability of B after event a has
occurred if you found this video helpful
consider supporting us on patreon to
help us make more videos you can also
visit our website at simple earning
procom to get access to many study
guides and practice questions thanks for
watching
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