Question

According to a daily​ newspaper, the probability is about 0.74 that the favorite in a horse race will finish in the money​ (first, second, or third​ place). Complete parts​ (a) through​ (j) below.

a. In the next five ​races, what is the probability that the favorite finishes in the money exactly​ twice? The probability that the favorite finishes in the money exactly twice is 0.096.

b. In the next five ​races, what is the probability that the favorite finishes in the money exactly four​ times? The probability that the favorite finishes in the money exactly four times is 0.390.

c. In the next five ​races, what is the probability that the favorite finishes in the money at least four​ times? The probability that the favorite finishes in the money at least four times is 0.611.

d. In the next five ​races, what is the probability that the favorite finishes in the money between two and four ​times, inclusive?

The probability that the favorite finishes in the money between two and four ​times, inclusive, is ____.

​(Round to three decimal places as​ needed.)

Answer 1

a)

Here, n = 5, p = 0.74, (1 - p) = 0.26 and x = 2
As per binomial distribution formula P(X = x) = nCx * p^x * (1 - p)^(n - x)

We need to calculate P(X = 2)
P(X = 2) = 5C2 * 0.74^2 * 0.26^3
P(X = 2) = 0.096

b)

Here, n = 5, p = 0.74, (1 - p) = 0.26 and x = 4
As per binomial distribution formula P(X = x) = nCx * p^x * (1 - p)^(n - x)

We need to calculate P(X = 4)
P(X = 4) = 5C4 * 0.74^4 * 0.26^1
P(X = 4) = 0.390

c)

Here, n = 5, p = 0.74, (1 - p) = 0.26 and x = 4
As per binomial distribution formula P(X = x) = nCx * p^x * (1 - p)^(n - x)

We need to calculate P(X >= 4).
P(X >= 4) = (5C4 * 0.74^4 * 0.26^1) + (5C5 * 0.74^5 * 0.26^0)
P(X >= 4) = 0.39 + 0.222
P(X >= 4) = 0.612

d)
Here, n = 5, p = 0.74, (1 - p) = 0.26, x1 = 2 and x2 = 4.
As per binomial distribution formula P(X = x) = nCx * p^x * (1 - p)^(n - x)

We need to calculate P(2 <= X <= 4)
P(2 <= X <= 4) = (5C2 * 0.74^2 * 0.26^3) + (5C3 * 0.74^3 * 0.26^2) + (5C4 * 0.74^4 * 0.26^1)
P(2 <= X <= 4) = 0.096 + 0.274 + 0.39
P(2 <= X <= 4) = 0.760