74% of all bald eagles survive their first year of life. Give your answers as decimals, not percents. If 33 bald eagles are randomly selected, find the probability that

Question

74% of all bald eagles survive their first year of life. Give your answers as decimals, not percents. If 33 bald eagles are randomly selected, find the probability that

a. Exactly 26 of them survive their first year of life. 

b. At most 26 of them survive their first year of life. 

c. At least 24 of them survive their first year of life. 

d. Between 19 and 24 (including 19 and 24) of them survive their first year of life. 

Answer 1

Hi Janessa,

This is a binomial case--i.e. two binary outcomes; in this case, first-year survival or not.

Formula for this:

P(X=x) = C(n, x) p^xq^n-x

n= Total number of birds in sample = 33

x= Desired number of survivors = 26 for part a; adjust accordingly for rest

p= Probability of Survival = 0.74

q= Probability of Death = 1 - p = 0.26

C(n,x) = n!/x!(n-x)!; !=Factorial=4*3*2*1, etc. You will need a calculator to input factorials.

So, for part a:

a.

n= 33

x= 26

n-x= 7

P(X=26) = C(33,26)0.74²⁶*0.26⁷

P (X=26) = 0.137

b. Question asks at most, so you have to account for 26 and everything below. You have two options. You can do 26 binomial computations as above or you can do 8 and apply the Complement Rule. I recommend the latter. You've already computed the probability of exactly 26, so you will need to compute the probabilities for 27 and higher. Example:

P(X=27) = C(33,27) 0.74²⁷*0.26⁶

P(X=28) = C(33,28) 0.74²⁸*0.26⁵

Repeat the procedure up to X=33 and add the probabilities together. This will give you the probability that more than 26 survive. You then apply the Complement Rule:

P(A^C) = 1 - P(A)

This means you subtract the probability you found above from 1 to get the probability that 26 survive at most.

c. Same procedure as part b, but start at 24, not 27. You will not need the Complement Rule here since the question asks for at least 24. Just add the probabilities together. Note that you can use the probabilities you obtained in parts a and b to help solve this part.

d. Again, same procedure.

P(X=19) = C(33,19) 0.74¹⁹*0.26¹⁴

P(X=20) = C(33,20) 0.74²⁰*0.26¹³

....

And so on down the line up to X=24. Then, add all the probabilities you found together to get the probability that between 19 and 24 birds survive the first year. I hope this helps.