A certain batch contains 5 % defectives defective. If 100 apples are randomly examined, what is

step 1 This question is about the binomial distribution here. So it is given a certain batch contains 5

A certain batch contains 5 % defectives defective. If 100...

Key Concepts

Cumulative Probability Calculation

Cumulative probability calculation involves finding the total probability of a random variable falling within a specific range, such as 'exceeding' or 'being less than' a set value, by summing individual probabilities or using a cumulative distribution function. This concept is essential in evaluating ranges of outcomes in probability distributions.

Continuity Correction

A continuity correction is used when approximating a discrete distribution, like the binomial, with a continuous one, such as the normal distribution. This involves adjusting the discrete variable by 0.5 units to better approximate the probability mass before or after a given cutoff, thus improving the accuracy of the normal approximation.

Binomial Distribution

The binomial distribution models the number of successes (or failures) in a fixed number of independent trials, with each trial having the same probability of success. It is applicable in situations such as determining the number of defective items in a batch, where each item has an independent chance of being defective.

Normal Approximation to the Binomial Distribution

When the number of trials is large and the success probability is not too close to 0 or 1, the binomial distribution can be approximated by a normal distribution. This approximation simplifies probability calculations for ranges of outcomes, especially when exact calculations become cumbersome.

Transcript

00:01 This question is about the binomial distribution here.

00:04 So it is given a certain batch contains 5 % defectives.

00:10 So that means the probability of getting a thing is 5%.

00:17 That means this is 5%, which is 0 .05.

00:22 And if 100 apples are randomly examined.

00:26 So in this case, the number of elements here, which is the 100 apples.

00:31 I'm going to just take as 100 here.

00:34 And what is the probability that the number of defective apples exceeds 15? first of all, since this is a binomial distribution, i'm going to just define the random variable x here, and this is binomial distributed.

00:49 And the probability is 0 .05, and the number of elements or apple here is 100.

00:55 So in part a, we have to just find the probability, of x is greater than which is 15.

01:04 So first of all, let's recall how we can just find the probability for binomial distribution.

01:11 Let me just write the formula here.

01:14 So the probability of this event which is equal to this is nx combination times p to the power x times q to the power and minus x and be careful the q means this is 1 minus p great.

01:30 So the next thing thing so we have to just use this formula here in order to get the answer here and what is given here so in this case the x is given as because we have to find the probability of x is greater than 15 so let me just write as this one so the probability of x is greater than 15 which is equal to that is the probability of x is equal to 16 plus the probability of x is equal to 17 up to because the since there's a 100, that means that goes to 100.

02:06 So it is very hard, you know, that calculate each of the calculation here.

02:11 What is supposed to do, we can just write this expression as x is greater than or equal to 16.

02:16 So in order to just find this answer, we have to use the binom cdf.

02:22 So i'm going to just use the binom cdf here.

02:25 So in order to get the probability of this one, let me just write here, the probability of x is greater than 16.

02:31 So the total probability, which is one minus, which is not included here.

02:37 The 15 and the list numbers are not included in this case.

02:42 That means we have to just subtract this probability.

02:45 So this is one minus...