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Question

Four fair dice are thrown independently 27 times.

Then the expected number of times, at least two dice show up as a three or a five, is


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Solution

Explanation for the correct option:

Step 1. Probability of getting 3 or 5:

Probability of getting 3or 5 (p)=26

=13

Probability of not showing 3or 5 (q)=1-13

=23

Step 2. Find the expected number of times :

Use the binomial distribution formula:

P(x)=Cnpxxqn-x

Where,

n=Total number of event

x=Total number of successful events

p=Probability of success

q=Probability of failure

The experiment is performed with four dices independently.

Step 3. Find the probability of showing 3or 5 at least twice in one throw of each dice is

C44p4+C34qp3+C24q2p2=1×p4+4×qp3+6×q2p2

Putting the value of p and q

134+4×23×133+6×232×132=181+881+2481=3381

Such experiment was performed 27 times

So expected outcome

μ=3381×27=11

Hence, the expected number of times, at least two dice show up as a three or a five is 11.


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