Question

You are running a race. The probability that you win is 3/5. What is the probability that you lose at most 2 out of your 6 races?

Answer 1

The probability that you lose at most 2 out of your 6 races is 0.54432.

Explanation:

We are given that you are running a race. The probability that you win is 3/5.

There are total of 6 races.

The above situation can be represented through binomial distribution;

`P(X = r) = \binom{n}{r} * p^(r) * (1-p)^(n-r) ; x = 0,1,2,3,.....`

where, n = number of trials (samples) taken = 6 races

r = number of success = at most 2 lost

p = probability of success which in our question is probability that

you lose a race = 1 - (3/5) = 2/5 or 0.4

Let X = Number of races lost

So, X ~ Binom(n = 6, p = 0.40)

Now, the probability that you lose at most 2 out of your 6 races is given by = P(X
`\leq` 2)

P(X
`\leq` 2) = P(X = 0) + P(X = 1) + P(X = 2)

=
`\binom{6}{0} * 0.40^(0) * (1-0.40)^(6-0) + \binom{6}{1} * 0.40^(1) * (1-0.40)^(6-1) + \binom{6}{2} * 0.40^(2) * (1-0.40)^(6-2)`

=
`1 *1 * 0.60^(6) + 6 * 0.40^(1) * 0.60^(5) +15* 0.40^(2) * 0.60^(4)`

= 0.54432