cam hits the bullseye in 8 darts out of 15 throws. what is the experimental probability that came next throws will hit the bullseye?

A darts player practices throwing a dart at the bull’s eye on a dart board. Her probability of hitting the bull’s eye for each throw is 0.2.

(a) Find the probability that she is successful for the first time on the third throw:

The number F of unsuccessful throws till the first bull’s eye follows a geometric

distribution with probability of success q = 0.2 and probability of failure p = 0.8.

If the first bull’s eye is on the third throw, there must be two failures:

P(F = 2) = p

2

q = (0.8)2

(0.2) = 0.128.

(b) Find the probability that she will have at least three failures before her first

success.

We want the probability of F ≥ 3. This can be found in two ways:

P(F ≥ 3) = P(F = 3) + P(F = 4) + P(F = 5) + P(F = 6) + . . .

= p

3

q + p

4

q + p

5

q + p

6

q + . . . (geometric series with ratio p)

=

p

3

q

1 − p

=

(0.8)3

(0.2)

1 − 0.8

= (0.8)3 = 0.512.

Alternatively,

P(F ≥ 3) = 1 − (P(F = 0) + P(F = 1) + P(F = 2))

= 1 − (q + pq + p

2

q)

= 1 − (0.2)(1 + 0.8 + (0.8)2

)

= 1 − 0.488 = 0.512.

(c) How many throws on average will fail before she hits bull’s eye?

Since p = 0.8 and q = 0.2, the expected number of failures before the first success

is

E[F] = p

q

=

0.8

0.2

= 4.