Handedness A psychologist needs 12 left-handed subjects for...

Handedness A psychologist needs 12 left-handed subjects for an experiment, and she interviews 15 potential subjects. About 10$\%$ of the population is left-handed. (a) What is the probability that exactly 12 of the potential subjects are left-handed? (b) What is the probability that 12 or more are left-handed?

Handedness A psychologist needs 12 left-handed subjects
for an experiment, and she interviews 15 potential subjects.
About 10$\%$ of the population is left-handed.
(a) What is the probability that exactly 12 of the potential
subjects are left-handed?
(b) What is the probability that 12 or more are left-handed?

Key Concepts

Probability Mass Function (PMF)

The probability mass function for the binomial distribution gives the probability of obtaining exactly k successes in n independent Bernoulli trials. It is expressed as P(X = k) = C(n, k) * p^k * (1-p)^(n-k), where p is the probability of success in each trial.

Cumulative Distribution and Complement Rule

Cumulative probabilities sum the probabilities of a random variable taking on a range of values. The complement rule is often used alongside, where the probability of obtaining at least a certain number of successes is found by subtracting the probability of fewer successes from one. This approach simplifies calculations in cumulative probabilistic analyses.

Binomial Distribution

The binomial distribution is a probability distribution that models the number of successes in a fixed number of independent trials, each with the same probability of success. It is used when each trial has only two possible outcomes, often termed 'success' and 'failure'.

Combination Formula

The combination formula, denoted as C(n, k) or 'n choose k', is used to calculate the number of ways to choose k items from a set of n items without regard to the order of selection. It is critical in determining the number of distinct outcomes in binomial probability computations.

Transcript

00:05 Okay, so in this problem, a psychologist needs 12 left -handed subjects for an experiment.

00:11 So we have, and she interviews 15 potential subjects with about 10 % of the population being left -handed.

00:20 So she'll have an n of 15, and p equals 0 .1.

00:32 So for part a, we're being asked, what is the probability that exactly 12 of the percentage? potential subjects are left -handed.

00:42 So for exactly 12, pr equals 12, we will have 15 choose 12, which looks like this, times our probability of success, which is 0 .1 to the 12th.

01:01 Which having 0 .1 to the 12th power already, this is not going to be a huge probability here.

01:07 And then our chances of failure are going to be 0 .9 cubed.

01:13 And so first step is, we want to do our 15, choose 12, and then we'll times 0 .1 to the 12th power, then times 0 .9 to the cubed power.

01:30 And that will give us this terribly tiny number, 3 .317 times 10 to the negative 10th power.

01:45 So this is not very good.

01:49 They should have done this differently.

01:54 But anyways, so now we've got the probability of r equals 12 being that number there.

01:59 And now we're going to look for what is the probability of 12 or more of them are left -handed.

02:07 So now we're doing p is r greater than or equal to 12.

02:13 And so then if our probability of r -equaling 12 was that small, we will get a slightly bigger number, but it is not going to be a huge number...

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