(Requires material from Section 3.3.) Refer to Example 4.14. Estimate the probability that exactly one of the four tires has a flaw, and find the uncertainty in the estimate.

Solution 25E

Step1 of 3:

Let us consider a random variable X it presents the number of tires out of four that have no flaw.

Also from Example 4.14 we have sample proportion and standard deviation Let “q” be the probability that exactly one of four tires has a flaw.

Then X follows binomial distribution with parameters “n and p” that is X B(n, p),

The probability mass function of binomial distribution is given by

, x = 0,1,2,...,n.

Where,

n = sample size

= 4

x = random variable

p = probability of success

q = 1 - p (probability of failure)

We need to Estimate the probability that exactly one of the four tires has a flaw, and also we need to find the uncertainty in the estimate.

Step2 of 3:

Consider,

Let q = P(X = 3)

Now,

=

= 4

= 4-4

Estimate of q is and it is given by

= 4 - 4

= 4(0.8043) - 4(0.7480)

= 3.2172 - 2.9920

= 0.2252

Hence = 0.2252.

Step3 of 3:

Consider,

Differentiate above equation with respect “” we get

= 12-16

= 12

= 12(0.8649) - 16(0.8043)

= 10.3788 - 12.8688

= -2.49

Hence, = -2.49

Therefore, the uncertainty in the estimate is given by

=

= (2.49)0.0255

= 0.0634

Hence the probability is 0.22520.0634.