Products from a certain machine are too large 15% of the time. What is the probability that in a run of 20 parts, 5 are too large?

Answer 1

#C_5^20(.15)^5(.85)^15~~.103#

The probability that 5 parts are too large is #.15xx.15xx.15xx.15xx.15=(.15)^5# and since 5 parts are too large, the other 15 are of acceptable size with probability #(1-.15)^15#. Since any 5 parts can be too large in no particular order, there are #C_5^20# ways to do this, so the probability that exactly 5 parts are too large is #C_5^20(.15)^5(.85)^15~~.103#.
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Answer 2

To find the probability that exactly 5 out of 20 parts are too large, you can use the binomial probability formula:

[ P(X = k) = \binom{n}{k} \times p^k \times (1 - p)^{n - k} ]

Where:

  • ( P(X = k) ) is the probability of getting exactly ( k ) successes,
  • ( n ) is the number of trials (in this case, 20 parts),
  • ( k ) is the number of successes (in this case, 5 parts being too large),
  • ( p ) is the probability of success on each trial (in this case, 15% or 0.15).

Plug in the values:

[ P(X = 5) = \binom{20}{5} \times 0.15^5 \times (1 - 0.15)^{20 - 5} ]

Calculate:

[ P(X = 5) = \binom{20}{5} \times 0.15^5 \times 0.85^{15} ]

[ P(X = 5) = \frac{20!}{5!(20-5)!} \times 0.15^5 \times 0.85^{15} ]

[ P(X = 5) = \frac{20!}{5!15!} \times 0.15^5 \times 0.85^{15} ]

[ P(X = 5) ≈ 0.226 ]

So, the probability that exactly 5 out of 20 parts are too large is approximately 0.226 or 22.6%.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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