Question

question 1 of 9 \\?\
biologists estimate that a randomly selected baby elk has a 44% probability of surviving to adulthood. assume this estimate is correct. suppose researchers choose 7 baby elk at random to monitor. let ( x = ) the number who survive to adulthood.\
find the probability that fewer than 3 of the elk survive to adulthood.\
(round to 4 decimal places. leave your answer in decimal form.)

Explanation:

This is a binomial probability problem where we want to find the probability that fewer than 3 out of 7 baby elk survive to adulthood. The probability of success (surviving) is \( p = 0.44 \), and the number of trials \( n = 7 \). We need to calculate \( P(X = 0) + P(X = 1) + P(X = 2) \) using the binomial probability formula \( P(X = k) = \binom{n}{k} p^k (1 - p)^{n - k} \).

Step 1: Calculate \( P(X = 0) \)

The binomial coefficient \( \binom{7}{0} = 1 \) (since \( \binom{n}{0} = 1 \) for any \( n \)).
\( p = 0.44 \), so \( p^0 = 1 \).
\( 1 - p = 0.56 \), and \( (1 - p)^{7 - 0} = 0.56^7 \).
\( P(X = 0) = \binom{7}{0} (0.44)^0 (0.56)^7 = 1 \times 1 \times 0.56^7 \)
Calculate \( 0.56^7 \approx 0.0194 \)
So \( P(X = 0) \approx 0.0194 \)

Step 2: Calculate \( P(X = 1) \)

The binomial coefficient \( \binom{7}{1} = \frac{7!}{1!(7 - 1)!} = \frac{7!}{1!6!} = 7 \)
\( p^1 = 0.44 \)
\( (1 - p)^{7 - 1} = 0.56^6 \)
\( P(X = 1) = \binom{7}{1} (0.44)^1 (0.56)^6 = 7 \times 0.44 \times 0.56^6 \)
Calculate \( 0.56^6 \approx 0.0346 \)
Then \( 7 \times 0.44 \times 0.0346 \approx 7 \times 0.0152 = 0.1064 \)

Step 3: Calculate \( P(X = 2) \)

The binomial coefficient \( \binom{7}{2} = \frac{7!}{2!(7 - 2)!} = \frac{7 \times 6}{2 \times 1} = 21 \)
\( p^2 = 0.44^2 = 0.1936 \)
\( (1 - p)^{7 - 2} = 0.56^5 \)
Calculate \( 0.56^5 \approx 0.0618 \)
\( P(X = 2) = \binom{7}{2} (0.44)^2 (0.56)^5 = 21 \times 0.1936 \times 0.0618 \)
First, \( 21 \times 0.1936 = 4.0656 \)
Then \( 4.0656 \times 0.0618 \approx 0.2513 \)

Step 4: Sum the probabilities

Now we sum \( P(X = 0) + P(X = 1) + P(X = 2) \)
\( 0.0194 + 0.1064 + 0.2513 = 0.3771 \)

Answer:

\( 0.3771 \)