In: Statistics and Probability
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)
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