A5. In testing a truck tyre over rugged terrain, it is found that of the trucks fail
to complete the test run without a blowout. Of the next 15 trucks tested, find the
probability that:
(a) from 3 to 6 inclusive have blow-outs
(b) fewer than 4 have blow-outs.
© more than 5 have blow-outs.

Question

A5. In testing a truck tyre over rugged terrain, it is found that

of the trucks fail
to complete the test run without a blowout. Of the next 15 trucks tested, find the
probability that:
(a) from 3 to 6 inclusive have blow-outs
(b) fewer than 4 have blow-outs.
© more than 5 have blow-outs.

UpStudy AI · Accepted Answer

Let $$ X $$ be the number of trucks that have blow-outs. Since each truck has a probability $$ p=0.25 $$ of having a blow-out and there are $$ n=15 $$ trucks tested, $$ X $$ follows a binomial distribution:
$$
X \sim \text{Binomial}(n=15, p=0.25)
$$
Thus, the general probability mass function is:
$$
P(X=k) = \binom{15}{k} (0.25)^k (0.75)^{15-k}
$$

### (a) Probability that from 3 to 6 trucks (inclusive) have blow-outs

We need to compute:
$$
P(3 \leq X \leq 6) = \sum_{k=3}^{6} \binom{15}{k} (0.25)^k (0.75)^{15-k}
$$

### (b) Probability that fewer than 4 trucks have blow-outs

This is equivalent to:
$$
P(X < 4) = P(X \leq 3) = \sum_{k=0}^{3} \binom{15}{k} (0.25)^k (0.75)^{15-k}
$$

### (c) Probability that more than 5 trucks have blow-outs

This probability is given by:
$$
P(X > 5) = 1 - P(X \leq 5) = 1 - \sum_{k=0}^{5} \binom{15}{k} (0.25)^k (0.75)^{15-k}
$$

These sums provide the required probabilities for parts (a), (b), and (c).
To find the probabilities: (a) The probability that between 3 and 6 trucks (inclusive) have blow-outs is calculated by summing the probabilities for 3, 4, 5, and 6 blow-outs. (b) The probability that fewer than 4 trucks have blow-outs is the sum of probabilities for 0, 1, 2, and 3 blow-outs. (c) The probability that more than 5 trucks have blow-outs is found by subtracting the sum of probabilities for 0 to 5 blow-outs from 1. These calculations use the binomial distribution with $$ n=15 $$ trials and a success probability of $$ p=0.25 $$ for each trial.

A5. In testing a truck tyre over rugged terrain, it is found that of the trucks fail
to complete the test run without a blowout. Of the next 15 trucks tested, find the
probability that:
(a) from 3 to 6 inclusive have blow-outs
(b) fewer than 4 have blow-outs.
© more than 5 have blow-outs.

Upstudy AI Solution

Answer

Solution

Bonus Knowledge

(a) Probability that from 3 to 6 inclusive have blow-outs

(b) Probability that fewer than 4 have blow-outs

© Probability that more than 5 have blow-outs

Summary of Probabilities

Related Questions

Latest Probability Questions

A5. In testing a truck tyre over rugged terrain, it is found that of the trucks fail to complete the test run without a blowout. Of the next 15 trucks tested, find the probability that: (a) from 3 to 6 inclusive have blow-outs (b) fewer than 4 have blow-outs. © more than 5 have blow-outs.

Upstudy AI Solution

Answer

Solution

Bonus Knowledge

(a) Probability that from 3 to 6 inclusive have blow-outs

(b) Probability that fewer than 4 have blow-outs

© Probability that more than 5 have blow-outs

Summary of Probabilities

Related Questions

Latest Probability Questions

A5. In testing a truck tyre over rugged terrain, it is found that of the trucks fail
to complete the test run without a blowout. Of the next 15 trucks tested, find the
probability that:
(a) from 3 to 6 inclusive have blow-outs
(b) fewer than 4 have blow-outs.
© more than 5 have blow-outs.