Question

Detective Decker knows that her partner, Mr. Morningstar, can successfully coerce a person into sharing their deepest, darkest secrets 85% of the time. Suppose they interrogate 35 random criminals at the precinct.

a) Which distribution does this scenario represent? Explain your answer.

b) Find the probability that exactly 15 criminals will share their deepest, darkest secrets.

c) Find the probability that at least 23 criminals will share their deepest, darkest secrets.

Answer 1

Probability that a criminal will share their deepest ,darkest secrets = 85% = 0.85

a) This scenario can be represented by Binomial distribution .

   We define a random variable X = no. of criminal who share their secrets out of 35

   Each criminal can be considered as a trial

   This X can have Binomial distribution because:

   (i) No. of trials or no. of criminals are constant = n = 35

   (ii) Probability of success(success is defined as a criminal sharing their secret) for one criminal is not dependent    on other criminals .

   (iii) There are only two possible outcomes that is , either the criminal shares or does not share .

   p = Probability of success on a single trial = 0.85

   So X~ B(35 , 0.85)

b) Probability distribution for a binomial distributionn is given as :

   P(X=x) gives probability of x no. of criminals sharing secret

(here is the combination operator )

So Probability that exactly 15 criminals will share their deepest , darkest secret = P(X=15)

c) Probability that atleast 23 will share their secrets = = P(X=23) + P(X=24) + -----------+ P(X=35)

  

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