Conditional probability and the multiplication rule

Июль 27, 2022

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Conditional probability and the multiplication rule

Содержание

2. Multiplication Rule The Multiplication Rule can be used to find the probability of two or more
3. Multiplication Rule Tip Find the probability the first event occurs. Find the probability the second event
4. Using the Multiplication Rule to find Probability A coin is tossed and a die is rolled.
5. Using the Multiplication Rule to find Probability A card is drawn from a deck and replaced;
6. Using the Multiplication Rule to find Probability The probability that a salmon swims successfully through a
7. Using the Multiplication Rule to find Probability Two cards are selected from a standard deck without
8. Using the Multiplication Rule to find Probability A Harris poll found the 46% of Americans say
9. Using the Multiplication Rule to find Probability The probability that a salmon swims successfully through a
10. Using the Multiplication Rule to find Probability Find the probability that none of the three salmon
11. Using the Multiplication Rule to find Probability Find the probability that at least one of the
12. Dependent Events When the outcome or occurrence of the first event affects the outcome or occurrence
13. Conditional Probability To find probabilities when events are dependent, use the multiplication rule with a modification
14. Finding Conditional Probability Two cards are selected in sequence from a standard deck. Find the probability
15. Finding Conditional Probability Three cards are drawn from an ordinary deck and not replaced. Find the
16. Finding Conditional Probability The table at the left shows the results of a study in which
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Multiplication Rule
The Multiplication Rule can be used to find the probability

of two or more events that occur in a sequence .
The multiplication Rule for the probability of A and B
If events A and B are independent, then the rule can be simplified to P(A and B) = P (A) ● P (B). This simplified rule can be extended for any number of independent events.

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Multiplication Rule Tip
Find the probability the first event occurs.
Find the probability

the second event occurs given the first event has occurred and
Multiply these two probabilities.

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Using the Multiplication Rule to find Probability
A coin is tossed

and a die is rolled. Find the probability of getting a head and then rolling a 6.

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Using the Multiplication Rule to find Probability
A card is drawn

from a deck and replaced; then a second card is drawn. Find the probability of selecting a Ace and then selecting a queen.

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Using the Multiplication Rule to find Probability
The probability that a

salmon swims successfully through a dam is 0.85. Find the probability that two salmon successfully swim through the dam.

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Using the Multiplication Rule to find Probability
Two cards are selected

from a standard deck without replacement. Find the probability that both are hearts.

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Using the Multiplication Rule to find Probability
A Harris poll found

the 46% of Americans say they suffer great stress at least once a week. If three people are selected at random, find the probability that all three will say they suffer great stress at least once a week.

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Using the Multiplication Rule to find Probability
The probability that a

salmon swims successfully through a dam is 0.85. Find the probability that three salmon swim successfully through the dam.

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Using the Multiplication Rule to find Probability
Find the probability that

none of the three salmon are successful.

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Using the Multiplication Rule to find Probability
Find the probability that

at least one of the three salmon is successful in swimming through the dam.

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Dependent Events
When the outcome or occurrence of the first event affects

the outcome or occurrence of the second event in such a way that the probability is changed, the events are said to be dependent events.
Examples
Drawing a card from a deck, NOT replacing it, and then drawing a second card.
Being a lifeguard and getting a tan.
Having high grades and getting a scholarship

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Conditional Probability
To find probabilities when events are dependent, use the multiplication

rule with a modification in notation.
P(A and B) = P (A) ● P (B\A).

The probability of B given that event A has already occured

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Finding Conditional Probability
Two cards are selected in sequence from

a standard deck. Find the probability that the second card is a queen, given that the first card is a king. (Assume that the king is not replaced)
Solution: Because the first card is a king and is not replaced, the remaining deck has 51 cards, 4 of which are queens. So,
P(B|A) = 4/51 = 0.078

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Finding Conditional Probability
Three cards are drawn from an ordinary

deck and not replaced. Find the probability of these events.
Getting 3 Jacks
Getting an ace, a king, and a queen in order
Getting a club, a spade, and a heart in order
Getting three clubs

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Finding Conditional Probability
The table at the left shows the results of

a study in which researchers examined a child’s IQ and the presence of a specific gene in the child. Find the probability that a child has a high IQ given that the child has the gene.
Solution: There are 72 children who have the gene. So, the sample space consists of these 72 children, as shown at the left. Of theses, 33 have a high IQ. So,
P(B\A)= 33/72 =0.458