Question

The probability is about 0.67 that the favorite horse race will finish in the money(1^st,2^nd,3^rd). In the next 5 races what is the probability that the favorite finishes in the money  

1. Exactly twice

2. Exactly 4 times

3. At least 4 times
4. Between 2 and 4 times, inclusive?

Answer 1

1)
Here, n = 5, p = 0.67, (1 - p) = 0.33 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 * 0.67^2 * 0.33^3
P(X = 2) = 0.1613

2)
We need to calculate P(X = 4)
P(X = 4) = 5C4 * 0.67^4 * 0.33^1
P(X = 4) = 0.3325

3)
We need to calculate P(X >= 4).
P(X >= 4) = (5C4 * 0.67^4 * 0.33^1) + (5C5 * 0.67^5 * 0.33^0)
P(X >= 4) = 0.3325 + 0.135
P(X >= 4) = 0.4675

4)
We need to calculate P(2 <= X <= 4)
P(2 <= X <= 4) = (5C2 * 0.67^2 * 0.33^3) + (5C3 * 0.67^3 * 0.33^2) + (5C4 * 0.67^4 * 0.33^1)
P(2 <= X <= 4) = 0.1613 + 0.3275 + 0.3325
P(2 <= X <= 4) = 0.8213