Answer:
a) 0.4133
b) 0.8724
c) 0.6429
d) yes, they are independent
Explanation:
Given 9 green cards and 5 yellow cards with events G1 = the first card drawn is green, and G2 = the second card drawn is green (after replacement of the first card), you want to know ...
- P(G1&G2)
- P(G1 or G2)
- P(G1|G2)
- whether the events are independent
a. Both green
There are 9 green cards among the 14 cards, so the probability of drawing a green one is 9/14. The drawings are done with replacement, so the second drawing has the same probability distribution as the first:
P(G1) = P(G2) = 9/14 . . . . . . . we also call this P(G)
The probability of both events occurring is the product of their individual probabilities:
P(G1&G2) = (9/14)(9/14) = 81/196
P(G1&G2) ≈ 0.4133
b. Either green
The probability of at least one green is the complement of the probability that none are green:
P(G1+G2) = 1 -((1 -P(G1))(1 -P(G2))) = P(G1) +P(G2) -P(G1)P(G2)
P(G1+G2) = P(G)·(2 -P(G)) = (9/14)(2 -9/14) = 171/196
P(G1+G2) ≈ 0.8724
c. Conditional probability
The conditional probability P(G2|G1) is found using the formula ...
P(G2|G1) = P(G1&G2)/P(G1) = P(G)²/P(G) = P(G) = 9/14
P(G2|G1) ≈ 0.6429
d. Independence
We know the events G1 and G2 are independent two ways:
- drawing is done with replacement, so the conditions for the second drawing are the same as the first. There is no interaction between the drawings (by definition).
- P(G2|G1) = P(G2), again indicating G2 is not dependent on G1.
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