Bill entered three contests. The probability he will win contest A is .2, the probability that he will win Contest B is .1 and the probability that he will win the Contest C is .6. Assuming that these events are all independent of one another, what is the probability that Bill will only win Contest B (that is, he will win Contest B but will not win Contest A nor Contest C)?
I'm guessing that should be written as
P (B∩Ccomplement∩Acomplement)? For complements in this case do you subtract from one? 1- p(b) in this case?
so you want A, notB, notC
= (.2)(.9)(.4)
=
To find the probability that Bill will only win Contest B, we can use the concept of independent events and the complement rule.
Let's break down the problem step by step:
1. The probability that Bill will win Contest A is given as P(A) = 0.2.
2. The probability that Bill will win Contest B is given as P(B) = 0.1.
3. The probability that Bill will win Contest C is given as P(C) = 0.6.
Now, we need to calculate the probability that Bill will win Contest B (B) but not win Contest A (A) nor Contest C (C). Expressing this mathematically, we have P(B ∩ A̅ ∩ C̅).
To calculate this probability, we can use the complement rule. For an event X, the complement rule states that P(X̅) = 1 - P(X), where X̅ represents the complement of event X.
Using this rule, we have:
P(B ∩ A̅ ∩ C̅) = P(B) × P(A̅) × P(C̅)
Since the events A, B, and C are assumed to be independent, we can multiply their probabilities together:
P(B) × P(A̅) × P(C̅) = P(B) × (1 - P(A)) × (1 - P(C))
Plugging in the given probabilities:
P(B ∩ A̅ ∩ C̅) = 0.1 × (1 - 0.2) × (1 - 0.6)
Simplifying:
P(B ∩ A̅ ∩ C̅) = 0.1 × 0.8 × 0.4 = 0.032
Therefore, the probability that Bill will only win Contest B is 0.032, or 3.2%.