Question

Sixty percent of dogs from a certain breed will chase a thrown ball. In a group of 15 dogs of this breed,

What’s the probability that less than 10 will chase a ball?

What’s the probability that at least 12 will chase a ball?

What’s the probability that exactly 8 will chase a ball?

What’s the probability that at most 4 will not chase a ball?

Answer 1

a)

Here, n = 15, p = 0.6, (1 - p) = 0.4 and x = 10
As per binomial distribution formula P(X = x) = nCx * p^x * (1 - p)^(n - x)

We need to calculate P(X < 10).
P(X < 10) = (15C0 * 0.6^0 * 0.4^15) + (15C1 * 0.6^1 * 0.4^14) + (15C2 * 0.6^2 * 0.4^13) + (15C3 * 0.6^3 * 0.4^12) + (15C4 * 0.6^4 * 0.4^11) + (15C5 * 0.6^5 * 0.4^10) + (15C6 * 0.6^6 * 0.4^9) + (15C7 * 0.6^7 * 0.4^8) + (15C8 * 0.6^8 * 0.4^7) + (15C9 * 0.6^9 * 0.4^6)
P(X < 10) = 0 + 0 + 0 + 0.002 + 0.007 + 0.024 + 0.061 + 0.118 + 0.177 + 0.207
P(X < 10) = 0.5968

b)

Here, n = 15, p = 0.6, (1 - p) = 0.4 and x = 12
As per binomial distribution formula P(X = x) = nCx * p^x * (1 - p)^(n - x)

We need to calculate P(X > =12).
P(X <= 11) = (15C0 * 0.6^0 * 0.4^15) + (15C1 * 0.6^1 * 0.4^14) + (15C2 * 0.6^2 * 0.4^13) + (15C3 * 0.6^3 * 0.4^12) + (15C4 * 0.6^4 * 0.4^11) + (15C5 * 0.6^5 * 0.4^10) + (15C6 * 0.6^6 * 0.4^9) + (15C7 * 0.6^7 * 0.4^8) + (15C8 * 0.6^8 * 0.4^7) + (15C9 * 0.6^9 * 0.4^6) + (15C10 * 0.6^10 * 0.4^5) + (15C11 * 0.6^11 * 0.4^4)
P(X <= 11) = 0 + 0 + 0.0003 + 0.0016 + 0.0074 + 0.0245 + 0.0612 + 0.1181 + 0.1771 + 0.2066 + 0.1859 + 0.1268

P(x< 11) = 0.9095

P(x> =12) = 1 - P(x< = 11)
= 1- 0.9095
= 0.0905

c)

Here, n = 15, p = 0.6, (1 - p) = 0.4 and x = 8
As per binomial distribution formula P(X = x) = nCx * p^x * (1 - p)^(n - x)

We need to calculate P(X = 8)
P(X = 8) = 15C8 * 0.6^8 * 0.4^7
P(X = 8) = 0.1771

d)

Here, n = 15, p = 0.6, (1 - p) = 0.4 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) = (15C0 * 0.6^0 * 0.4^15) + (15C1 * 0.6^1 * 0.4^14) + (15C2 * 0.6^2 * 0.4^13) + (15C3 * 0.6^3 * 0.4^12) + (15C4 * 0.6^4 * 0.4^11)
P(X <= 4) = 0 + 0 + 0.0003 + 0.0016 + 0.0074
P(X <= 4) = 0.0093