A fair coin is tossed four times, and a person win ₹ 1 for each head and lose ₹ 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

Solution:

Since the coin is tossed four times, there can be a maximum of 4 heads and tails.

• When 4 heads turn up, ₹(1 + 1 + 1 + 1) = 4, i.e., ₹ 4 is the gain.
• When 3 heads and 1 tail turn up, ₹(1 + 1 + 1 - 1.50) = 3 -1.50 = 1.50, i.e., ₹1.50 is the gain.
• When 2 heads and 2 tails turn up, ₹(1 + 1 - 1.50 - 1.50) = 2 - 3 = - 1, i.e., ₹ 1 is the loss.
• When 1 head and 3 tails turn up, ₹(1 - 1.50 - 1.50 - 1.50) = 1 - 4.50 = - 3.50 , i.e., ₹ 3.50 is the loss.
• When 4 tails turn up, ₹(- 1.50 - 1.50 - 1.50 - 1.50) = - 6 , i.e., ₹6 is the loss.

There are 2⁴ = 16 elements in the sample space S, which is given by:

S = {HHHH, HHHT, HHTH, HTHH, THHH, HHTT, HTTH, TTHH, HTHT, THTH, THHT, HTTT, THTT, TTHT, TTTH, TTTT}

Therefore, n (S) = 16. Now we will find the probabilities of each of the above gain/loss.

• The person wins Rs 4.00 when 4 heads turn up, i.e., when the event {HHHH} occurs.
  Hence, Probability (winning ₹ 4) = 1/16.
   
• The person wins ₹1.50 when 3 heads and 1 tail turn up, i.e., when the event {HHHT, HHTH, HTHH, THHH} occurs.
  Hence, Probability (winning ₹1.50 ) = 4/16 = 1/4.
   
• The person loses ₹ 1 when 2 heads and 2 tails turn up, i.e., when the event {HHTT, HTTH, TTHH, HTHT, THTH, THHT} occurs.
  Hence, Probability (losing ₹1) = 6/16 = 3/8.
   
• The person loses ₹ 3.50 when 1 head and 3 tails turn up, i.e., when the event {HTTT, THTT, TTHT, TTTH} occurs.
  Hence, Probability (losing ₹ 3.50 ) = 4/16 = 1/4.
   
• The person losses ₹ 6 when 4 tails turn up, i.e., when the event {TTTT} occurs.
  Hence, Probability (losing ₹ 6) = 1/16

NCERT Solutions Class 11 Maths Chapter 16 Exercise 16.3 Question 7

A fair coin is tossed four times, and a person win ₹ 1 for each head and lose ₹ 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.

Summary:

Probability (winning ₹ 4) = 1/16, Probability (winning ₹1.50) = 1/4, Probability (losing ₹1) = 3/8, Probability (losing ₹ 3.50) = 1/4, Probability (losing ₹ 6) = 1/16