Conditional probability and the multiplication rule презентация

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.

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

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

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

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

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

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.

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

none of the three salmon are successful.

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

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

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

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

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

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

the gene.
Find the probability that a child does not have the gene, given that the child has a normal IQ.