Question

The homicide clearance rate in a certain state is 65​%.A crime is cleared when an arrest is​ made, a crime is​ charged, and the case is referred to a court.

a. What is the probability that exactly 6 out of 10 independent homicides are​ cleared? The probability that exactly 6 out of 10 independent homicides are cleared

is Answer ______. ​(Type an integer or decimal. Round to three decimal places as​ needed.)

b. Without doing a​ calculation, state whether the probability that 6 or more out of 10 homicides are cleared will be larger or smaller than the answer to part​ (a). Why? The probability that 6 or more out of 10 homicides are cleared is larger than the answer to part​ (a) because the probability that 6 or more out of 10 homicides are cleared includes the probability that exactly 6 homicides are cleared.

c. What is the probability that 6 or more out of 10 independent homicides are​ cleared? The probability that 6 or more out of 10 independent homicides are cleared

is Answer ________ (Type an integer or decimal. Round to three decimal places as​ needed.)

Answer 1

a)
Here, n = 10, p = 0.65, (1 - p) = 0.35 and x = 6
As per binomial distribution formula P(X = x) = nCx * p^x * (1 - p)^(n - x)

We need to calculate P(X = 6)
P(X = 6) = 10C6 * 0.65^6 * 0.35^4
P(X = 6) = 0.238

b)
The probability that 6 or more out of 10 homicides are cleared is larger than the answer to part​ (a) because the probability that 6 or more out of 10 homicides are cleared includes the probability that exactly 6 homicides are cleared.

c)
We need to calculate P(X >= 6).
P(X >= 6) = (10C6 * 0.65^6 * 0.35^4) + (10C7 * 0.65^7 * 0.35^3) + (10C8 * 0.65^8 * 0.35^2) + (10C9 * 0.65^9 * 0.35^1) + (10C10 * 0.65^10 * 0.35^0)
P(X >= 6) = 0.238 + 0.252 + 0.176 + 0.072 + 0.013
P(X >= 6) = 0.751