Each weld passes inspection with probability 0.90 independently. In a lot of 8 welds, what is the probability exactly 2 fail?

Start with the thinking

• Define success as the event you are counting — here, failure.
• Use the binomial pmf, not the normal approximation, for n = 8.

Step-by-step solution

1. Binomial pmf

   $$P(X = k) = C(n,k) p^k (1 - p)^(n-k)$$

2. Combinations

   $$C(8,2) = 8!/(2!6!) = 28$$

3. Probability terms

   $$(0.10)^{2} = 0.0100 and (0.90)^{6} = 0.5314$$

4. Substitute

   $$P = 28(0.0100)(0.5314)$$

5. Evaluate

   $$P = 0.149$$

Why the other options are there

• 0.0100 (combinations omitted)
• 0.383 (P(X ≤ 2) computed)

Reference: FE Reference Handbook — Probability and Statistics — Binomial distribution

A probability and statistics problem uses z-score. Given mean (mu) = 5,390 psi; std deviation (sigma) = 140.0 psi; value (x) = 3,020 psi, determine the z-score (z).

Start with the thinking

• The governing relation printed in this handbook section is z-score.
• Everything except z is given, so isolate z symbolically first — never rearrange after the numbers are in.
• Tabulate each given with its unit and confirm the units are consistent with the relation before substituting.
• Probability and Statistics items reward recognising the unknown before touching a calculator.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$z = (x - \mu) / \sigma$$

2. Step 2 — Rearrange the relation so that z stands alone on the left-hand side.

3. Step 3

   $$List the givens: mean (mu) = 5,390 psi, std deviation (sigma) = 140.0 psi, value (x) = 3,020 psi$$

4. Step 4 — Substitute the given values into the rearranged relation.

5. Step 5 — Evaluate:

   $$z = -16.9286$$

6. Step 6 — Check: returning z = -16.9286 to

   $$z = (x - \mu) / \sigma$$

   reproduces the given quantities, and both sides carry the same units.

Why the other options are there

• -33.8571 — kept a factor of two that cancels in the correct rearrangement.
• -8.4643 — dropped that same factor in the other direction.
• -18.6214 — rounded an intermediate value before the final step.

Reference: FE Reference Handbook — Probability and Statistics → Binomial Distribution

A probability and statistics problem uses Binomial mean. Given trials (n) = 100.0; success probability (p) = 0.3100, determine the mean (mu).

Start with the thinking

• The governing relation printed in this handbook section is Binomial mean.
• Everything except mu is given, so isolate mu symbolically first — never rearrange after the numbers are in.
• Tabulate each given with its unit and confirm the units are consistent with the relation before substituting.
• Probability and Statistics items reward recognising the unknown before touching a calculator.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$\mu = n p$$

2. Step 2 — Rearrange the relation so that mu stands alone on the left-hand side.

3. Step 3

   $$List the givens: trials (n) = 100.0, success probability (p) = 0.3100$$

4. Step 4 — Substitute the given values into the rearranged relation.

5. Step 5 — Evaluate:

   $$\mu = 31.0000$$

6. Step 6 — Check: returning mu = 31.0000 to

   $$\mu = n p$$

   reproduces the given quantities, and both sides carry the same units.

Why the other options are there

• 62.0000 — kept a factor of two that cancels in the correct rearrangement.
• 15.5000 — dropped that same factor in the other direction.
• 34.1000 — rounded an intermediate value before the final step.

Reference: FE Reference Handbook — Probability and Statistics → Binomial Distribution

The binomial distribution predicts the number of failures in a batch of tests. Given number of trials (n) = 100.0; success probability (p) = 0.2700; variance (var) = 16.9200, determine the mean (mu).

Start with the thinking

• The governing relation printed in this handbook section is Binomial distribution mean and variance.
• Everything except mu is given, so isolate mu symbolically first — never rearrange after the numbers are in.
• Tabulate each given with its unit and confirm the units are consistent with the relation before substituting.
• The binomial distribution describes the number of successes in n independent trials with probability p.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$\mu = n p, \quad \sigma^2 = n p (1-p)$$

2. Step 2 — Rearrange symbolically for mu:

   $$\mu = np$$

3. Step 3 — List the givens: number of trials (n) = 100.0, success probability (p) = 0.2700, variance (var) = 16.9200.

4. Step 4 — Substitute the given values:

   $$\mu = n0.2700$$

5. Step 5 — Evaluate:

   $$\mu = 27.0000$$

6. Step 6 — Check: returning mu = 27.0000 to

   $$\mu = n p, \quad \sigma^2 = n p (1-p)$$

   reproduces the given quantities, and both sides carry the same units.

Why the other options are there

• 54.0000 — kept a factor of two that cancels in the correct rearrangement.
• 13.5000 — dropped that same factor in the other direction.
• 29.7000 — rounded an intermediate value before the final step.

Reference: FE Handbook — Binomial

A probability and statistics problem uses z-score. Given mean (mu) = 4,500 psi; std deviation (sigma) = 400.0 psi; z-score (z) = -1.6600, determine the value (x) in psi.

Start with the thinking

• The governing relation printed in this handbook section is z-score.
• Everything except x is given, so isolate x symbolically first — never rearrange after the numbers are in.
• Tabulate each given with its unit and confirm the units are consistent with the relation before substituting.
• Probability and Statistics items reward recognising the unknown before touching a calculator.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$z = (x - \mu) / \sigma$$

2. Step 2 — Rearrange the relation so that x stands alone on the left-hand side.

3. Step 3

   $$List the givens: mean (mu) = 4,500 psi, std deviation (sigma) = 400.0 psi, z-score (z) = -1.6600$$

4. Step 4 — Substitute the given values into the rearranged relation.

5. Step 5 — Evaluate:

   $$x = 3836\ \text{psi}$$

6. Step 6 — Check: returning x = 3,836 psi to

   $$z = (x - \mu) / \sigma$$

   reproduces the given quantities, and both sides carry the same units.

Why the other options are there

• 7,672 — kept a factor of two that cancels in the correct rearrangement.
• 1,918 — dropped that same factor in the other direction.
• 4,220 — rounded an intermediate value before the final step.

Reference: FE Reference Handbook — Probability and Statistics → Binomial Distribution

A probability and statistics problem uses Binomial mean. Given success probability (p) = 0.2800; mean (mu) = 60.5000, determine the trials (n).

Start with the thinking

• The governing relation printed in this handbook section is Binomial mean.
• Everything except n is given, so isolate n symbolically first — never rearrange after the numbers are in.
• Tabulate each given with its unit and confirm the units are consistent with the relation before substituting.
• Probability and Statistics items reward recognising the unknown before touching a calculator.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$\mu = n p$$

2. Step 2 — Rearrange the relation so that n stands alone on the left-hand side.

3. Step 3

   $$List the givens: success probability (p) = 0.2800, mean (mu) = 60.5000$$

4. Step 4 — Substitute the given values into the rearranged relation.

5. Step 5 — Evaluate:

   $$n = 216.1$$

6. Step 6 — Check: returning n = 216.1 to

   $$\mu = n p$$

   reproduces the given quantities, and both sides carry the same units.

Why the other options are there

• 432.1 — kept a factor of two that cancels in the correct rearrangement.
• 108.0 — dropped that same factor in the other direction.
• 237.7 — rounded an intermediate value before the final step.

Reference: FE Reference Handbook — Probability and Statistics → Binomial Distribution

An engineer models the binomial distribution of defective parts in a lot. Given number of trials (n) = 134.0; success probability (p) = 0.8200; mean (mu) = 32.5000, determine the variance (var).

Start with the thinking

• The governing relation printed in this handbook section is Binomial distribution mean and variance.
• Everything except var is given, so isolate var symbolically first — never rearrange after the numbers are in.
• Tabulate each given with its unit and confirm the units are consistent with the relation before substituting.
• The binomial distribution describes the number of successes in n independent trials with probability p.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$\mu = n p, \quad \sigma^2 = n p (1-p)$$

2. Step 2 — Rearrange symbolically for var:

   $$var = np(1-p)$$

3. Step 3 — List the givens: number of trials (n) = 134.0, success probability (p) = 0.8200, mean (mu) = 32.5000.

4. Step 4 — Substitute the given values:

   $$var = n0.8200(1-0.8200)$$

5. Step 5 — Evaluate:

   $$var = 19.7784$$

6. Step 6 — Check: returning var = 19.7784 to

   $$\mu = n p, \quad \sigma^2 = n p (1-p)$$

   reproduces the given quantities, and both sides carry the same units.

Why the other options are there

• 39.5568 — kept a factor of two that cancels in the correct rearrangement.
• 9.8892 — dropped that same factor in the other direction.
• 21.7562 — rounded an intermediate value before the final step.

Reference: FE Handbook — Binomial

A probability and statistics problem uses z-score. Given std deviation (sigma) = 230.0 psi; value (x) = 4,000 psi; z-score (z) = -1.3100, determine the mean (mu) in psi.

Start with the thinking

• The governing relation printed in this handbook section is z-score.
• Everything except mu is given, so isolate mu symbolically first — never rearrange after the numbers are in.
• Tabulate each given with its unit and confirm the units are consistent with the relation before substituting.
• Probability and Statistics items reward recognising the unknown before touching a calculator.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$z = (x - \mu) / \sigma$$

2. Step 2 — Rearrange the relation so that mu stands alone on the left-hand side.

3. Step 3

   $$List the givens: std deviation (sigma) = 230.0 psi, value (x) = 4,000 psi, z-score (z) = -1.3100$$

4. Step 4 — Substitute the given values into the rearranged relation.

5. Step 5 — Evaluate:

   $$\mu = 4301\ \text{psi}$$

6. Step 6 — Check: returning mu = 4,301 psi to

   $$z = (x - \mu) / \sigma$$

   reproduces the given quantities, and both sides carry the same units.

Why the other options are there

• 8,603 — kept a factor of two that cancels in the correct rearrangement.
• 2,151 — dropped that same factor in the other direction.
• 4,731 — rounded an intermediate value before the final step.

Reference: FE Reference Handbook — Probability and Statistics → Binomial Distribution

A probability and statistics problem uses Binomial mean. Given trials (n) = 168.0; mean (mu) = 6.1000, determine the success probability (p).

Start with the thinking

• The governing relation printed in this handbook section is Binomial mean.
• Everything except p is given, so isolate p symbolically first — never rearrange after the numbers are in.
• Tabulate each given with its unit and confirm the units are consistent with the relation before substituting.
• Probability and Statistics items reward recognising the unknown before touching a calculator.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$\mu = n p$$

2. Step 2 — Rearrange the relation so that p stands alone on the left-hand side.

3. Step 3

   $$List the givens: trials (n) = 168.0, mean (mu) = 6.1000$$

4. Step 4 — Substitute the given values into the rearranged relation.

5. Step 5 — Evaluate:

   $$p = 0.0363$$

6. Step 6 — Check: returning p = 0.0363 to

   $$\mu = n p$$

   reproduces the given quantities, and both sides carry the same units.

Why the other options are there

• 0.0726 — kept a factor of two that cancels in the correct rearrangement.
• 0.0182 — dropped that same factor in the other direction.
• 0.0399 — rounded an intermediate value before the final step.

Reference: FE Reference Handbook — Probability and Statistics → Binomial Distribution

A student computes the mean and variance of a binomial distribution. Given success probability (p) = 0.1800; mean (mu) = 126.9; variance (var) = 1.8700, determine the number of trials (n).

Start with the thinking

• The governing relation printed in this handbook section is Binomial distribution mean and variance.
• Everything except n is given, so isolate n symbolically first — never rearrange after the numbers are in.
• Tabulate each given with its unit and confirm the units are consistent with the relation before substituting.
• The binomial distribution describes the number of successes in n independent trials with probability p.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$\mu = n p, \quad \sigma^2 = n p (1-p)$$

2. Step 2 — Rearrange symbolically for n:

   $$n = \dfrac{\mu}{p}$$

3. Step 3

   $$List the givens: success probability (p) = 0.1800, mean (mu) = 126.9, variance (var) = 1.8700$$

4. Step 4 — Substitute the given values:

   $$n = \dfrac{126.9}{0.1800}$$

5. Step 5 — Evaluate:

   $$n = 705.0$$

6. Step 6 — Check: returning n = 705.0 to

   $$\mu = n p, \quad \sigma^2 = n p (1-p)$$

   reproduces the given quantities, and both sides carry the same units.

Why the other options are there

• 1,410 — kept a factor of two that cancels in the correct rearrangement.
• 352.5 — dropped that same factor in the other direction.
• 775.5 — rounded an intermediate value before the final step.

Reference: FE Handbook — Binomial