Question

Biologists estimate that a randomly selected baby elk has a 44% chance of surviving to adulthood. Assume this estimate is

correct. Suppose researchers choose 7 baby elk at random to monitor. Let X = the number that survives to adulthood.

What is P(X = 4)? Interpret this value.


A) (4) (0.44)* (0.56)'. There is a 23.04% probability that at least 4 of the 7 elk survive to adulthood.

B) (4) (0.44) (0.56)*. There is a 23.04% probability that exactly 4 of the 7 elk survive to adulthood.

C) (0.44)* (0.56)*. There is a 0.66% probability that at least 4 of the 7 elk survive to adulthood.

D) (0.44)* (0.56)'. There is a 0.66% probability that exactly 4 of the 7 elk survive to adulthood.

E) This probability cannot be calculated because this is not a binomial setting.

Answer 1

The probability of exactly 4 out of 7 baby elks surviving to adulthood is approximately 23.3%. This is calculated using the binomial probability formula, specifically for P(X = 4) given the survival probability of 44%.

Step-by-step explanation:

The question given is related to binomial probability distribution, which is a common topic in high school mathematics courses. To compute P(X = 4) where X is the number of baby elks surviving to adulthood, we use the binomial probability formula:

P(X = k) = C(n, k) * pk * (1-p)(n-k)

Where:

• C(n, k) is the combination of n items taken k at a time.
• n is the number of trials (in this case, 7 baby elks).
• k is the number of successes (in this case, 4 baby elks).
• p is the probability of success on any given trial (44% chance of survival, or 0.44).

Using this formula:

P(X = 4) = C(7, 4) * 0.444 * (1-0.44)(7-4)

Calculating this gives:

P(X = 4) = 35 * 0.444 * 0.563

P(X = 4) = 35 * (0.0377) * (0.1756)

P(X = 4) ≈ 0.2330 or 23.3%

Therefore, there is a 23.3% probability that exactly 4 of the 7 baby elks will survive to adulthood. The correct answer is option B, which slightly misstated the probability value as 23.04%, but correctly interprets the type of probability as 'exactly' rather than 'at least'.