Question

A study found that 25% of car owners in Fiji had their cars washed professionally rather than do it themselves. If 18 car owners are randomly selected, find the probability that at most two people have their cars washed professionally.

Answer 1

The probability that at most two people have their cars washed professionally is 0.1353.

Step-by-step explanation:

We are given that a study found that 25% of car owners in Fiji had their cars washed professionally rather than do it themselves.

18 car owners are randomly selected.

The above situation can be represented through binomial distribution;

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

where, n = number of trials (samples) taken = 18 cars

           r = number of success = at most two people wash cars professionally

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

                 car owners in Fiji wash cars professionally, i.e; p = 25%

Let X = Number of people have their cars washed professionally

So, X ~ Binom(n = 18 , p = 0.25)

Now, Probability that at most two people have their cars washed professionally is given by = P(X $\leq$ 2)

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

=  $\binom{18}{0} \times 0.25^{0} \times (1-0.25)^{18-0}+\binom{18}{1} \times 0.25^{1} \times (1-0.25)^{18-1}+\binom{18}{2} \times 0.25^{2} \times (1-0.25)^{18-2}$

=  $1 \times 1 \times 0.75^{18}+18 \times 0.25^{1} \times 0.75^{17}+153 \times 0.25^{2} \times 0.75^{16}$

=  0.1353

Therefore, probability that at most two people have their cars washed professionally is 0.1353.

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