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.
Step-by-Step Solution
Key Concept: Use the concept of equally likely outcomes. For independent tosses of a fair coin, each sequence of heads (H) and tails (T) is equally likely. Probability = (number of favourable outcomes) / (total number of outcomes).
1. Identify the sample space\\
For three tosses, each toss has two possible results (H or T). Hence the total number of possible outcomes is $$2^3 = 8.$$\\
The sample space \(S\) can be listed as:
$$S = \{HHH, HHT, HTH, HTT, THH, THT, TTH, TTT\}.$$\\
2. Determine the winning outcomes\\
Hanif wins only when all three tosses are the same, i.e., either \(HHH\) or \(TTT\). Thus the number of winning outcomes \(W\) is 2.\\
3. Determine the losing outcomes\\
The losing outcomes are the remaining sequences. Hence
$$\text{Number of losing outcomes} = 8 - 2 = 6.$$\\
4. Compute the probability of losing\\
Using the definition of probability for equally likely outcomes:
$$P(\text{lose}) = \frac{\text{Number of losing outcomes}}{\text{Total outcomes}} = \frac{6}{8} = \frac{3}{4}.$$\\
5. Answer\\
Therefore, the probability that Hanif will lose the game is $$\boxed{\frac{3}{4}}.$$
Correct Answer: 3/4