Suppose that a is the probability of success in an...

Suppose that $a$ is the probability of success in an experiment and $b=1-a$ is the probability of failure. If the experiment is repeated three times, show that the probabilities of the number of successes $S$ being $3,2,1,0$, respectively, are given by $P(S=3)=1 a^{3}, P(S=2)=3 a^{2} b, P(S=1)=$ $3 a b^{2}$, and $P(S=0)=1 b^{3}$.

Key Concepts

Probability Mass Function

The probability mass function (PMF) in the context of the binomial distribution gives the probability that a discrete random variable is exactly equal to a particular value. In this case, it is used to calculate the probability of obtaining exactly k successes in n independent trials, by multiplying the binomial coefficient by the appropriate powers of the success and failure probabilities.

Binomial Coefficient

The binomial coefficient, denoted as C(n, k) or 'n choose k', is a combinatorial tool that calculates the number of ways to choose k successes out of n trials. It is defined as n!/(k!(n-k)!) and plays a critical role in the binomial probability formula by accounting for the different arrangements in which the successes can occur.

Bernoulli Trial

A Bernoulli trial is a random experiment with exactly two possible outcomes, often labeled as “success” and “failure”. Each trial is characterized by the probability of success (often denoted by a parameter such as p) and the probability of failure (which is 1-p). These trials are the foundational building blocks for more complex probability models involving repeated trials.

Binomial Distribution

The binomial distribution describes the probability distribution of the number of successes in a fixed number of independent Bernoulli trials. With parameters n (number of trials) and p (probability of success on each trial), the distribution provides the likelihood of observing exactly k successes, encompassing various possible outcomes across all trials.

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