A game consists of tossing a one rupee coin 3 times and noting its outcome each time. Hanif wins if all the tosses give the same result i.e., three heads or three tails, and loses otherwise. Calculate the probability that Hanif will lose the game.
When a coin is tossed 3 times, the possible outcomes are: HHH, HHT, HTH, HTT, THH, THT, TTH, TTT. So, the total number of outcomes = 8. Hanif wins if all tosses give the same result, i.e., three heads (HHH) or three tails (TTT). So, number of outcomes favourable to winning = 2. Therefore, Hanif loses in the remaining outcomes. Number of outcomes favourable to losing = 8 – 2 = 6. Hence, P(Hanif will lose) = Number of favourable outcomes / Total number of outcomes = 6/8 = 3/4.
Explanation
This question from Exercise 14.1 tests understanding of sample space and probability calculation. Following the approach shown in Example 9 of the textbook, students must first enumerate all possible outcomes when tossing a coin 3 times. The key insight is that Hanif wins only in 2 specific cases (all heads or all tails), and loses in all other cases. The probability of losing can be calculated directly as 6/8 = 3/4.
Solution Steps
Step 1: List all possible outcomes when a coin is tossed 3 times: HHH, HHT, HTH, HTT, THH, THT, TTH, TTT. Total outcomes = 8.
Step 2: Identify winning outcomes (same result on all tosses): HHH and TTT. Number of winning outcomes = 2.
Step 3: Calculate losing outcomes = Total outcomes - Winning outcomes = 8 - 2 = 6.
Step 4: Calculate probability of losing = 6/8 = 3/4.