Final answer:
The probability of Luther choosing exactly 2 words with a Latin root out of 4 attempts can be determined by using the binomial probability formula, applying the given success probability (27%) and the number of trials (4).
Step-by-step explanation:
Luther faces a binomial distribution problem because he can either choose a word with a Latin root (defined as a success) or without (a failure) when selecting words to study for the Spelling Bee. With a probability of a word having a Latin root being 27% (or p = 0.27), and with 4 attempts to choose words (n = 4), we want to find the probability that exactly 2 of these words have a Latin root (X = 2 successes).
To calculate this, we will use the binomial probability formula:
P(X = x) = (n choose x) * p^x * q^(n-x)
where:
- n is the number of trials
- x is the number of successful trials we are interested in
- p is the probability of success on a single trial
- q is the probability of failure on a single trial (q = 1 - p)
In this case:
- n = 4 (the 4 times Luther chooses a word)
- x = 2 (we're looking for exactly 2 words with a Latin root)
- p = 0.27 (the probability a word has a Latin root)
- q = 0.73 (the probability a word does not have a Latin root, which is 1 - 0.27)
We calculate "4 choose 2" (which is 6), p^2 (which is 0.27²), and q^2 (which is 0.73²). The probability is therefore:
P(X = 2) = 6 * (0.27)^2 * (0.73)^2
After performing the calculation, we get Luther's probability of choosing exactly 2 words with a Latin root out of 4 attempts.