Download Section 7B: Combining Probabilities

Section 7B: Combining Probabilities
Section 7B: Combining Probabilities
Example. Suppose you roll a fair six-sided die twice. What is the probability that you will get a 2 on the
first roll, and then an odd number on the second roll?
And Probability: Independent Events
Two events are independent if the outcome of one does not affect the probability of the other event.
Consider two independent events A and B with individual probabilities P (A) and P (B). The probability
that A and B occur together is
P (A and B) = P (A) × P (B)
This principle can be extended to any number of independent events. For example, the probability of
three events A, B, and C all occurring is
P (A and B and C) = P (A) × P (B) × P (C)
Example.
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You flip a coin twice. What is the probability that
you will get heads both times?
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You flip a coin three times. What is the probability that you will get heads, then tails, then heads
again?
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You roll a fair six-sided die twice. What is the
probability that you will get snake-eyes (two 1’s)?
box-cars (two 6’s)?
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Being an avid minigolfer, the probability you
make a hole-in-one is 0.15. What is the probability that on your next two holes, you don’t make
a hole in one?
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Section 7B: Combining Probabilities
Example. There are 10 freshmen and 15 sophomores in a class, and two must be selected for the school
senate. If each person is equally likely to be selected, what is the probability that both people selected are
sophomores?
And Probability: Dependent Events
Two events are dependent if the outcome of one event affects the probability of the other event. The
probability that dependent events A and B occur together is
P (A and B) = P (A) × P (B given A)
where P (B given A) means “the probability of event B given the occurrence of event A.” This principle can be extended to any number of dependent events. For examples, the and probability of three
dependent events A, B, and C is
P (A and B and C) = P (A) × P (B given A) × P (C given A and B)
Example.
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In the example above, what is the probability that
those selected are both freshmen?
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Ten names are thrown in a hat for winning a raffle, eight women and two men. Three tickets are
drawn out. What is the probability that they will
all be women? all men?
Section 7B: Combining Probabilities
Example. You roll a standard fair six-sided die once. What is the probability that you will get a 4 or an
odd value?
Either/Or Probability: Non-Overlapping Events
Two events are non-overlapping if they cannot occur together. If A and B are non-overlapping events,
the probability that either A or B occurs is
P (A or B) = P (A) + P (B)
This principle can be extended to any number of non-overlapping events. For example, the probability
that either event A, event B, or event C occures is
P (A or B or C) = P (A) + P (B) + P (C)
Example.
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You flip a coin three times. What is the probability of getting all heads or all tails?
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You draw a card from a standard 52-card deck.
What is the probability the card will be a spade
or a heart?
Section 7B: Combining Probabilities
Example. You roll two standard fair six-sided dice. What is the probability that at least one of the dice
is a 2 (that is, the first die is a 2 or the second die is a two?)
Either/Or Probability: Overlapping Events
Two events are overlapping if they can occur together. If A and B are overlapping events, the probability
that either A or B occurs is
P (A or B) = P (A) + P (B) − P (A and B)
Example. Refer to the previous example. What is the probability that at least one die has an odd value?
You draw a card from a standard 52-card deck. What is the probability that you get a jack or a spade?
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Section 7B: Combining Probabilities
Example. You draw two cards from a standard 52-card deck. What is the probability that
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Both cards are red?
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Both cards are 10?
Example. You roll two dice. What is the probability that
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at least one is a multiple of 3?
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they sum to 7 or 11?
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their sum is even?
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their sum is odd?
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Section 7B: Combining Probabilities
Example. You flip an unfair coin three times. The probability of getting heads is
probability of getting all outcomes the same?
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.
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What is the
Example. A poll is given to a large sampling of students. The poll found that 40% of those sampled like
their math class. Additionally, it found that 55% like cheeseburgers.
I Assume these events are independent (that is, liking math and liking cheeseburgers are unrelated). What
is the probability that someone polled liked both math and cheeseburgers? liked math and disliked
cheeseburgers?
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Now, assume that these events are overlapping, and the percentage of those who liked both is 15%.
What is the probability that that someone polled would like neither math nor cheeseburgers.
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Now, assume these events are disjoint. What is the probability that a person polled likes both math
and cheeseburgers? Likes neither math nor cheeseburgers?
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