Question

Binomial Distribution : Last year, Luther came in 2nd place in his school's Spelling Bee. This year, he plans to win 1st place. He compiled a long list of words to study, 27% of which have a Latin root. If Luther randomly chooses a word to study from the list 4 different times this month, what is the probability that exactly 2 of the words have a Latin root?

Answer 1

To find the probability that exactly 2 out of 4 chosen words have a Latin root when 27% of words on Luther's list are Latin-rooted, use the binomial distribution formula with n=4 trials, p=0.27 probability of success, and q=0.73 probability of failure.

Step-by-step explanation:

The student's question is related to the concept of binomial distribution, which is a type of probability distribution in mathematics. Specifically, the question asks for the probability that exactly 2 out of 4 randomly chosen words from Luther's list of words, with 27% of them having a Latin root, will have a Latin root.

In order to solve this problem, the binomial probability formula can be used: P(X = x) = (n choose x) * p^x * q^(n-x), where 'P(X = x)' is the probability of exactly x successes in n trials, 'n choose x' is the combination of n items taken x at a time, 'p' is the probability of success on a single trial, and 'q' is the probability of failure on a single trial (which equals 1 - p).

In this case, we have the following:

• n = 4 (the number of trials)

• x = 2 (the desired number of successes)

• p = 0.27 (the probability of choosing a word with a Latin root)

• q = 0.73 (the probability of not choosing a word with a Latin root, which is 1 - 0.27)

Therefore, the probability that Luther will randomly choose exactly 2 words with a Latin root from his list can be calculated as follows:

P(X = 2) = (4 choose 2) * (0.27)^2 * (0.73)^2

After doing the calculations, the probability can be determined. This exercise helps Luther understand his chances of selecting a specific type of word during his study sessions.