Question

Suppose you flip one million coins. Would you be surprised to find that 501,000 landed heads? Would you be surprised to find that 510,000 landed heads? Explain.

Answer 1

Answer)

N = 1 million = 10,00000

P = 0.5 {as in a single toss, we have one heads and one tails}

So, equal probability of getting heads or tails = 0.5

First we need to check the conditions of normality that is if n*p and n*(1-p) both are greater than 5 or not

N*p = 5,00,000

N*(1-p) = 5,00,000

Both the conditions are met so we can use standard normal z table to estimate the probability

Z = (x - mean)/s.d

Mean = n*p = 5,00,000

S.d = √{n*p*(1-p)} = 500

P(501,000) = P(500999.5 < x < 501000.5)

= P(x<501000.5) - P(x<500999.5)

P(x<501000.5)

Z = (501000.5 - 500000)/500 = 2

From z table, P(z<2) = 0.9772

P(x<500999.5)

Z = (500999.5 - 500000)/500 = 2

From z table, P(z<2) = 0.9772

= 0.9772 - 0.9772

= 0

As the probability is extremely less we will be surprised

B)

P(510000) = p(509999.5<x<510000.5)

P(510000.5 - 500000)/500 = 20.

From z table, P(z<20) = 1

Z = (509999.5 - 500000)/500 = 20

From z table, P(z<20) = 1

Probability = 1 - 1 = 0

Here also probability is extremely small

So we will be surprised