Adult salmon have lengths that are normally distributed with a mean of m = 70 cm

Question

Adult salmon have lengths that are normally distributed with a mean of m = 70 cm and a standard deviation of s = 5.4 cm. Assume you go fishing with a friend and you each catch at least one adult salmon.

a. What is the probability that the first fish caught is no longer than 80 cm?

b. What is the probability that the second fish caught is between 68 and 74 cm in length?

c. What is the probability that the first adult salmon you caught is at least 8 cm longer than the first adult salmon your friend catches?

Explanation / Answer

Solution:

Let random variable X denote the length of adult salmon which are normally distributed X ~N(70, 5.42)

We know that if X ~ N(, 2) then the transformed random variable is, Z = X-/

a) Find the probability that the first fish caught is no longer than 80 cm

P(X 80) = 1-P(X 80)
= 1- P(8070/5.4)
= 1- P (Z >1.85)
= 1P ( Z<1.85 ) =10.9678 = 0.0322

b)  What is the probability that the second fish caught is between 68 and 74 cm in length

  P ( 68 < X < 74 ) = P ( 6870/5.4 < X/ < 7470/5.4)
=P ( 0.37 < Z < 0.74 )
= ( Z < 0.74 )P (Z < 0.37 )
= 0.7704 - 0.3557
= 0.4147

c) What is the probability that the first adult salmon you caught is at least 8 cm longer than the first adult salmon your friend catches

P(X1 -X2 8) = 1- P(X1-X2 8)
= 1- P(8-0/2*5.4)
= 1- P(8/7.6367)
= 1-P(1.047)
= 1-0.8524
= 0.1476

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