Question

A fair coin is tossed four times, and a person win Re 1 for each head and lose Rs 1.50 for each tail that turns up.

From the sample space calculate how many different amounts of money you can have after four tosses and the probability of having each of these amounts.

Answer 1

There are five ways in which a head can be obtained in a coin toss. These are as follows.

Total possible outcomes = {HHHH, HHHT, HHTH, HHTT, HTHH, HTHT, HTTH, HTTT, THHH, THHT, THTT, TTHH, TTHT, TTTH, TTTT}

(i) No head is obtained or all four tails are obtained.

Loss on getting all four tails = 4 × 1.50

= Rs. 6

Ways of getting four tails (TTTT) = 1

Total possible outcomes = 16

∴ Probability of getting four tails = `1/16`

(ii) When one head and 3 tails are obtained.

Loss = 3 × 1.50 – 1 × 1

= 4.50 – 1.00

= Rs. 3.50

A head and 3 tails can come up as follows:

{TTTH, TTHT, THTT, HTTT}

∴ A head and 3 tails can come up in 4 ways.

Total possible outcomes = 16

Probability of getting a head = `6/16`

= `1/4`

(iii) When 2 heads and 2 tails appear

Loss = 2 × 1.5 – 1 × 2

= 3 – 2

= Rs. 1

2 heads and 2 tails can come up as follows.

{HHTT, HTHT, HTTH, THHT, THTH, TTHH}

There are six ways in which 2 heads and 2 tails can be obtained.

Total possible outcomes = 16

Probability of getting 2 heads = 2

(iv) When 3 heads and 1 tail appear, then

Profit = 3 × 1 – 1 × 1.5

= 3 – 1.50

= Rs. 1.50

Ways of getting 3 heads = {HHHT, HHHH, HTHH, THHH}

There are four ways in which 3 heads and 1 tail can be obtained.

Total possible outcomes = 16

Probability of getting 3 heads = `4/16`

= `1/4`

(v) All four heads can be obtained in one way, then

Profit = 4 × 1

= Rs. 4

Total possible outcomes = 16

Probability of getting four heads = `4/16`

= `1/4`