Bulls Eye

Bulls Eye
Warriors snap 3-game skid by beating Bulls 114-97
Monta Ellis had 36 points, eight assists and five rebounds, Corey Maggette added 32 points, six rebounds and five assists and the short-handed Golden State Warriors snapped a three-game losing streak with a 114-97 victory over the Chicago Bulls on Monday. Stephen Curry had 26 points, 10 rebounds and six assists and Andris Biedrins grabbed a season-high 19 rebounds to go with a career-best eight...
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Given that the probability that an archer will hit the bulls eye at a distance of 50 feet is .6?

What is the probability that he hits the bullseye for the first time on his 3rd shot?
What is the expected number of arrows the archer will shoot before he hits the bulls eye?

Dear Anna,

For the archer to hit the bull's-eye for the first time on his third shot, he must miss it on his first two shots. If you believe that his chance of hitting the bull's-eye on any shot is always 0.6, independent of previous shots, then you can solve this with the geometric distribution.

The geometric distribution is often written as:
f(k) = (1 - p)^(k - 1) p, where p is the probability of success on each trial, and k is the trial on which the first success occurs, for k in {1, 2, 3, . . . }.
If k is not in {1, 2, 3, . . . } then f(k) = 0.

For your question p = 0.6 and k = 3, so
f(3) = (1 - 0.6)^(3 - 1)(0.6)
= (0.4)^2 (0.6)
= (0.16)(0.6)
= 0.096 .

The expected number of arrows shot to hit the bull's-eye the first time is simply 1 / p. To get the expected number of arrow shot BEFORE hitting the bull's-eye, you would need to subtract one shot (i. e., the shot that hits the bull's-eye). Thus, the expected number of arrows shot before the archer hits the bull's-eye the first time is
1 / p - 1
= (1 - p) / p
= (1 - 0.6) / 0.6
= 0.4 / 0.6
= 2 / 3
= 0.667 (to three decimal places).

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Many thanks for reading our Bulls Eye article

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