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Two events are said to be either dependent or independent. Here are two tests for independence.

Events A and B are independent if and only if P(A) = P(A | B).

That is, A is independent of B if and only if the probability of A happening is the same whether or not B has happened.

Events A and B are independent if and only if P(A)·P(B) = P(A and B).

Example:
Joe picks a card at random from a standand deck of cards. Let A = "it's a heart" and B = "it's a seven."

P(A) = 13/52. There are 13 hearts out of 52.
P(A | B) = 1/4. If it is known the card is a seven, the probability it's a heart is 1/4.

13/52 = 1/4. So P(A) = P(A | B). A and B are independent.

Also, let's check that P(A)·P(B) = P(A and B).
P(A) = 13/52 = 1/4
P(B) = 4/52 = 1/13 since there are 4 sevens in the entire deck.
P(A and B) = 1/52 since there is only one seven that is a heart.
1/4 · 1/13 = 1/52. So P(A)·P(B) = P(A and B). A and B are independent.

Example:
What is the probability that Joe will flip a coin 8 times in a row and all will be "heads"?

Since the outcome of each flip does not depend on what previously happened,
P(8 heads in a row)
= P(heads)· P(heads)· P(heads)· P(heads)· P(heads)· P(heads)· P(heads)· P(heads)
= (1/2)⁸ = 0.0039

Example:
If events A and B are independent, P(A) = 0.40 and P(B) = 0.85 then the following area model helps to see the probabilities of compound events.

P(A and B) = (0.40)(0.85) = 0.34
P(A and not B) = (0.40)(0.15) = 0.06
P(not A and B) = (0.60)(0.85) = 0.51
P(not A and not B) = (0.60)(0.15) = 0.09

Example:
Joe picks two cards–one right after the other without replacing the first. What is the probability that both are kings?

P(1st card king) = 4/52. P(2nd card king|1st card king) = 3/51.
P(both kings) = 4/52 * 3/51 = 12/2652 = 1/221 = .0045 = 0.45%
He's got less than 1% chance to pick 2 kings.
Note: the probability of picking the second king depended on whether or not he picked the 1st king.

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Unit 7 Review - Dependent and independent events

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Integrated 2 Teachers

Events A and B are independent if and only if P(A) = P(A | B).

Events A and B are independent if and only if P(A)·P(B) = P(A and B).

Example:	Joe picks a card at random from a standand deck of cards. Let A = "it's a heart" and B = "it's a seven." P(A) = 13/52. There are 13 hearts out of 52. P(A \| B) = 1/4. If it is known the card is a seven, the probability it's a heart is 1/4. 13/52 = 1/4. So P(A) = P(A \| B). A and B are independent. Also, let's check that P(A)·P(B) = P(A and B). P(A) = 13/52 = 1/4 P(B) = 4/52 = 1/13 since there are 4 sevens in the entire deck. P(A and B) = 1/52 since there is only one seven that is a heart. 1/4 · 1/13 = 1/52. So P(A)·P(B) = P(A and B). A and B are independent.

Example:	What is the probability that Joe will flip a coin 8 times in a row and all will be "heads"? Since the outcome of each flip does not depend on what previously happened, P(8 heads in a row) = P(heads)· P(heads)· P(heads)· P(heads)· P(heads)· P(heads)· P(heads)· P(heads) = (1/2)⁸ = 0.0039

Example:	If events A and B are independent, P(A) = 0.40 and P(B) = 0.85 then the following area model helps to see the probabilities of compound events. P(A and B) = (0.40)(0.85) = 0.34 P(A and not B) = (0.40)(0.15) = 0.06 P(not A and B) = (0.60)(0.85) = 0.51 P(not A and not B) = (0.60)(0.15) = 0.09

Example:	Joe picks two cards–one right after the other without replacing the first. What is the probability that both are kings? P(1st card king) = 4/52. P(2nd card king\|1st card king) = 3/51. P(both kings) = 4/52 * 3/51 = 12/2652 = 1/221 = .0045 = 0.45% He's got less than 1% chance to pick 2 kings. Note: the probability of picking the second king depended on whether or not he picked the 1st king.