Question

Suppose you have an experiment where you flip a coin three times. You then count the number of heads. a.) State the random variable. b.) Write the probability distribution for the number of heads. c.) Draw a histogram for the number of heads. d.) Find the mean number of heads. e.) Find the variance for the number of heads. f.) Find the standard deviation for the number of heads. g.) Find the probability of having two or more number of heads. h.) Is it unusual to flip two heads

Answer 1

Answer:

The answer to each point is below

Step-by-step explanation:

We will solve point by point:

a)  We have to:

Random variable X = number of heads

Let, H => heads, T => tails

b) We have that the combinations are

TTT, TTH, THT, THH, HTT, HTH, HHT, HHH

Number of Heads (X)  0     1      2     3

Probability (P)             1/8   3/8  3/8   1/8

c) attached the histogram.

d)  We have the following:

Mean = E (X) = 0 * 1/8 + 1 * 3/8 + 2 * 3/8 + 3 * 1/8

m = 1.5

The mean is 1.5

e) E (X ^ 2) = 0 * 1/8 + 12 * 3/8 + 22 * 3/8 + 32 * 1/8 = 3

Variance = E (X ^ 2) - (E (X)) ^ 2

Var = 3 - (1.5) ^ 2

Var = 0.75

The variance is 0.75

f) standard deviation = (Var) ^ (1/2) = (0.75) ^ (1/2) = 0.866

sd = 0.866

the standard deviation is 0.866

g) P (2 or more heads) = 3/8 + 1/8 = 0.5

The probability is 50%

h)  P (two heads) = 3/8 = 0.375

It is likely that out of 8 times of 3 flips, 3 times we can observe two heads out of 3, therefore it is not unusual.