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Two buddies, Zeke and Bubba, plan a squirrel-hunting trip. Bubba is a better shot than Zeke, specifically, Bubba is twice as good a shot as Zeke. Upon seeing a squirrel, Bubba and Zeke raise their rifles and simultaneously shoot at the squirrel. Assume that Bubba shooting the squirrel and Zeke shooting the squirrel are independent events, and that the probability of Zeke shooting the squirrel is 0.2.

Part (a) What is the probability that the squirrel will be shot?
\[ P(\text{squirrel shot}) = \] (use four decimals)

Part (b) What is the probability that both Zeke and Bubba will miss the squirrel?
\[ P(\text{both miss}) = \] (use four decimals)

Part (c) If the squirrel is shot, what is the probability that Zeke shot the squirrel?
\[ P(\text{Zeke shot the squirrel}) = \] (use four decimals)

Part (d) If only one of the two buddies shot the squirrel, what is the probability it was Bubba?
(use four decimals)

Answer :

The probability that the squirrel will be shot is 0.6.the probability that both Zeke and Bubba will miss the squirrel is 0.32.The probability that Zeke shot the squirrel is approximately 0.3333. The probability that if only one of the two buddies shot the squirrel, it was Bubba is 0.4

Given:

- Probability of Zeke shooting the squirrel: P(Zeke shoots) = 0.2

- Bubba is 2 times a better shot than Zeke

(a) To find the probability that the squirrel will be shot, we need to calculate the probability that either Zeke or Bubba will shoot the squirrel. Since the events are independent, we can use the probability of Zeke shooting the squirrel (P(Zeke shoots) = 0.2) and the fact that Bubba is 2 times a better shot.

P(squirrel shot) = P(Zeke shoots) + P(Bubba shoots)

P(squirrel shot) = 0.2 + 2 x 0.2 = 0.2 + 0.4 = 0.6

Therefore, the probability that the squirrel will be shot is 0.6.

(b) To find the probability that both Zeke and Bubba will miss the squirrel, we need to calculate the probability that neither Zeke nor Bubba shoot the squirrel. Since the events are independent, we can multiply their probabilities.

P(both miss) = P(Zeke does not shoot) x P(Bubba does not shoot)

P(both miss) = (1 - P(Zeke shoots)) x (1 - P(Bubba shoots))

P(both miss) = (1 - 0.2) x (1 - 0.6) = 0.8 x 0.4 = 0.32

Therefore, the probability that both Zeke and Bubba will miss the squirrel is 0.32.

(c) If the squirrel is shot, we need to find the probability that Zeke shot the squirrel. Since Bubba is 2 times a better shot, the probability that Bubba shot the squirrel is twice the probability of Zeke shooting the squirrel.

P(Zeke shot the squirrel) = P(Zeke shoots) / (P(Zeke shoots) + 2 x P(Zeke shoots))

P(Zeke shot the squirrel) = 0.2 / (0.2 + 2 x 0.2) = 0.2 / 0.6 = 0.3333

Therefore, the probability that Zeke shot the squirrel is approximately 0.3333.

(d) To find the probability that if only one of the two buddies shot the squirrel, it was Bubba, we need to calculate the conditional probability that Bubba shot the squirrel given that the squirrel was shot.

P(Bubba shot | squirrel shot) = P(Bubba shot) / P(squirrel shot)

P(Bubba shot | squirrel shot) = 2 x P(Zeke shoots) / P(squirrel shot)

P(Bubba shot | squirrel shot) = 2 x 0.2 / 0.6 = 0.4

Therefore, the probability that if only one of the two buddies shot the squirrel, it was Bubba is 0.4.

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