Question

A study found that 25% of car owners in UK 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

Step-by-step explanation:

Use the concept of binomial probability distribution.

Let the event of a car owner having their car washed professionally be counted as success.

Now according to Binomial Probability distribution, if under n tries, the probability of success is p and that of no success be q, then the probability of r successes is given by:

$P(X=r) = \binom{n}{r} p^r q^{n-r}$

So the probability of at most 2 successes is :

$P(X=0)+P(X=1)+P(X=2)=...\\=\binom{18}{0}(0.25)^0(0.75)^{18} + \binom{18}{1} (0.25)^1(0.75)^{17} + \binom{18}{2}(0.25)^2(0.75)^{16}$

Answer 2

Given :  25% of car owners in UK had their cars washed professionally rather than do it themselves. 18 car owners are randomly selected,

To find : the probability that at most two people have their cars washed professionally

Solution:

25% of car owners in UK had their cars washed professionally rather than do it themselves.

Probability cars washed professionally p = 25/100 = 1/4

Probability cars not washed professionally q = 1 - 1/4 = 3/4

P(x) =  ⁿCₓpˣqⁿ⁻ˣ

18 car owners s elected

=> n = 18

probability that at most two people have their cars washed professionally

0 people , 1 people , 2 people  have their cars washed professionally

=> P(0) + P(1) + P(2)

=   ¹⁸C₀(1/4)⁰(3/4)¹⁸  +  ¹⁸C₁(1/4)¹(3/4)¹⁷ +  ¹⁸C₂(1/4)²(3/4)¹⁶

=  (3¹⁶ / 4¹⁸ ) ( 9 +  54  +  153)

=  (3¹⁶ / 4¹⁸ ) (216)

=   24 * ( 3/4)¹⁸

= 0.1353

= 13.53 %  

13.53 % is the probability that at most two people have their cars washed professionally

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