Question

For each of the following questions, say whether the random process is a binomial process or not, and explain your answer. As part of your explanation, you will want to comment on the potential validity of each of the four things that must be true for a process to be a binomial process.

a. One basketball player attempts 10 free throws and the number of successful attempts is totalled.
b. Ten different basketball players each attempt 1 free throw and the total number of successful attempts is totalled.

Answer 1

Both scenarios a and b meet the conditions for a binomial process, with a fixed number of trials, two outcomes (success or failure), independence of trials, and constant probability of success.

Step-by-step explanation:

The question concerns whether certain scenarios represent a binomial process. A binomial experiment must satisfy four conditions: a fixed number of trials, only two possible outcomes (success or failure), trials must be independent, and the probability of success must remain constant throughout the trials.

a. A single basketball player attempting 10 free throws is a binomial process because:

• The number of trials (free throws) is fixed at 10.
• Each trial has only two outcomes: a success (making the shot) or a failure (missing the shot).
• The trials are independent; the outcome of one free throw doesn't affect the others.
• The probability of making a free throw (success) remains constant for each attempt.

b. Ten different basketball players each attempting 1 free throw is also a binomial process since:

• There is a fixed number of trials (each player making one attempt).
• There are only two possible outcomes for each free throw.
• The free throw attempts are independent of each other.
• The probability of each player making the shot is assumed constant, though it could vary between players; however, each player gets only one attempt, so this condition is effectively met.

Answer 2

a

This process follows is a binomial process

b

This process is not a binomial process

Step-by-step explanation:

Considering the first statement

One basketball player attempts 10 free throws and the number of successful attempts is totaled.

This process is a binomial process because

1. The number of trials is finite
2. Each trial is independent of the other
3. There is only two possible outcome (make a basket and make no basket
4. The probability of making a basket and not making a basket is constant throughout the trials

One basketball player attempts 10 free throws and the number of successful attempts is totaled.

This process is not a binomial process because chance that is the probability of success and failure will remain constant throughout the trial is not equal (what this means is that given the number of player is more than on their probability of success and failure will differ.)