On average 1 in 10 of the chocolates produced in a factory are mis-shapes.In a random sample of 1000 chocolates,find the probability that i) fewer than 80 are mis-shapes.

Question

On average 1 in 10 of the chocolates produced in a factory are mis-shapes.In a random sample of 1000 chocolates,find the probability that i) fewer than 80 are mis-shapes.

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Answer 1

The number of unacceptable chocolate bars in a sample of 1000

1000
is a random variable 𝑋
X
which has the binomial distribution with parameters 𝑛=1000
n=1000
and 𝑝=0.10
𝑋∼Bin(𝑛=1000,𝑝=0.10)
i.
To calculate the required probabilities we will use the normal approximation to the binomial distribution, that is
𝑋∼N(𝜇=𝑛𝑝,σ^2=𝑛𝑝(1−𝑝))
X∼N(μ=np,σ^2=np(1−p))
approximately. Here μ=𝑛𝑝=1000⋅0.1=100
μ=np=1000⋅0.1=100
and σ2=𝑛𝑝(1−𝑝)=100⋅0.9=90
σ^2=np(1−p)=100⋅0.9=90
. Applying also the continuity correction we have that

For A)
𝑃(𝑋<80)=𝑃(𝑋≤79)≈𝑃(𝑋−μσ≤79+1/2−10090‾‾‾√)=𝑃(𝑍≤−2.16)==Φ(−2.16)=1−Φ(2.16)=0.015
P(X<80) =P(X≤79)≈P(X−μσ≤79+1/2−10090)
=P(Z≤−2.16)=
Φ(−2.16)=1−Φ(2.16)=0.015
using the normal distribution tables.

Answer 2

To find the probability that fewer than 80 chocolates are misshapes out of a random sample of 1000 chocolates, we can use the binomial probability formula.

The binomial probability formula is given by:

P(X=k) = (nCk) * p^k * (1-p)^(n-k)

where:
- P(X=k) is the probability of getting exactly k successes
- n is the total number of trials or samples
- k is the number of successes
- p is the probability of success in a single trial
- (nCk) is the number of combinations of n items taken k at a time

In this case, the probability of a chocolate being a misshape (success) is 1/10, since 1 in 10 chocolates are misshapes.

Let's calculate the probability of getting fewer than 80 misshapes out of 1000 chocolates:

P(X < 80) = P(X=0) + P(X=1) + P(X=2) + ... + P(X=79)

Using the binomial probability formula, we can calculate each individual probability and sum them up.

P(X < 80) = ∑ (nCk) * p^k * (1-p)^(n-k) from k=0 to 79

It's a lengthy calculation, but if you'd like, I can calculate it step-by-step for you.

Answer 3

To find the probability that fewer than 80 chocolates are mis-shapes in a random sample of 1000 chocolates, we can use the binomial probability formula. The binomial probability formula is given by:

P(x) = (nCx) * p^x * (1-p)^(n-x)

Where:
- P(x) is the probability of getting exactly x successes
- n is the total number of trials or samples taken (in this case, 1000)
- x is the number of successes (in this case, the number of mis-shapes chocolates, which is less than 80)
- p is the probability of success (in this case, the probability that a chocolate is mis-shape, which is 1/10 or 0.1)
- (nCx) is the number of combinations of n items taken x at a time (also known as the binomial coefficient)

Now, let's calculate the probability using this formula.

First, let's find the probability of getting 79 or fewer mis-shapes chocolates:
P(X ≤ 79) = P(X = 0) + P(X = 1) + P(X = 2) + ... + P(X = 79)

Using the binomial probability formula, we can calculate each term and sum them up.

P(X = 0) = (1000C0) * (0.1)^0 * (1-0.1)^(1000-0)
P(X = 1) = (1000C1) * (0.1)^1 * (1-0.1)^(1000-1)
...
P(X = 79) = (1000C79) * (0.1)^79 * (1-0.1)^(1000-79)

Finally, we can sum up all these probabilities to get the desired result:

P(X ≤ 79) = P(X = 0) + P(X = 1) + P(X = 2) + ... + P(X = 79)

Note that this calculation involves a large number of terms, so using a calculator or statistical software would be helpful.