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.
👍
To find the probability that fewer than 80 chocolates are misshapes in a random sample of 1000 chocolates, we can use the binomial distribution formula.
The binomial distribution formula is given by:
P(X = k) = C(n, k) * p^k * (1-p)^(n-k),
where:
P(X = k) is the probability of having exactly k successes,
C(n, k) is the number of combinations of n items taken k at a time (also known as the binomial coefficient),
p is the probability of success (in this case, the probability of a chocolate being a misshape), and
n is the number of trials (in this case, the total number of chocolates in the sample).
To find the probability that fewer than 80 chocolates are misshapes, we need to calculate the sum of the probabilities for all values less than 80. Therefore, we need to calculate P(X < 80).
P(X < 80) = P(X = 0) + P(X = 1) + P(X = 2) + ... + P(X = 79).
Given that 1 in 10 chocolates is a misshape, the probability of a chocolate being a misshape (p) is 1/10, or 0.1. Therefore, the probability of a chocolate not being a misshape (1-p) is 1 - 0.1, or 0.9.
Now, we can substitute these values into the binomial distribution formula to calculate the probability for each value of k, from 0 to 79, and add them together to get the final probability.
Note: Calculating the probability for each value individually can be time-consuming. However, it can be facilitated by using statistical software, a calculator, or even programming languages that offer binomial distribution functions.
To find the probability that fewer than 80 chocolates are mis-shapes in a random sample of 1000 chocolates, we can use the binomial distribution formula.
The probability of getting exactly "k" successes (mis-shapes) in "n" trials (chocolates) is given by the formula:
P(X = k) = (n choose k) * p^k * (1-p)^(n-k)
Where:
- (n choose k) is the binomial coefficient, which calculates the number of ways to choose "k" successes from "n" trials.
- p is the probability of success (mis-shape) on a single trial.
- (1-p) is the probability of failure (correct shape) on a single trial.
Given that 1 in 10 chocolates are mis-shapes, the probability of a single chocolate being a mis-shape is p = 1/10 = 0.1. The probability of a chocolate being correctly shaped is (1-p) = 1 - 0.1 = 0.9.
Since we want to find the probability of fewer than 80 mis-shapes, we need to calculate:
P(X < 80) = P(X = 0) + P(X = 1) + P(X = 2) + ... + P(X = 79)
To calculate this, we can use a cumulative distribution function (CDF) table for the binomial distribution, or we can use software, such as Excel or a statistical calculator. The calculation can be quite tedious by hand.
Using an Excel spreadsheet or equivalent software, the probability can be calculated by summing the individual probabilities for each possible number of mis-shapes from 0 to 79.
The probability will be the sum of P(X = 0), P(X = 1), P(X = 2), ..., P(X = 79).
Note: Since calculating this by hand can be time-consuming, it is recommended to use software for such operations.