Hi please help me with the statistic problem below, make sure write down detaile

Question

Hi please help me with the statistic problem below, make sure write down detailed steps for each part, thank you!

1) A gill net survey conducted in the 1990s in a certain lake found that 55% of fish in the lake were cutthroat trout. Last year catch and release data from anglers provide species information for a total of 12 fish caught in the lake. 1.What is the probability that a) All 12 fish will be cutthroat trout b) Exactly 5 fish will be cutthroat trout c) Less than 5 fish will be cutthroat trout d Five of more fish will be cutthroat trout

Explanation / Answer

X ~ Binomial (n,p)

Where n = 12, p = 0.55

The binomial probability distribution is

P(X) = nCx px (1-p)n-x

a)

P( X = 12) = 12C12 0.5512 0.450

= 0.000766

b)

P( X = 5) = 12C5 0.555 0.457

= 0.148945

c)

P( X < 5) = P( X <= 4)

= P( X = 0) + P( X = 1) + P( X = 2) + P( X = 3) + P( X = 4)

= 12C0 0.550 0.4512 +12C1 0.55 0.4511 +12C2 0.552 0.4510 +12C3 0.553 0.459 +12C4 0.554 0.458

= 0.11174

d)

P( X >= 5) = 1 - P( X < 5)

= 1 - 0.11174

= 0.88826

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