A game consists of tossing a one rupee coin three 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.

Options

A. `(1)/(4)`
B. `(1)/(2)`
C. `(3)/(4)`
D. `(5)/(8)`

Answer

To solve the problem of calculating the probability that Hanif will lose the game when tossing a coin three times, we can follow these steps:

### Step 1: Determine the total number of possible outcomes
When tossing a coin, there are 2 possible outcomes for each toss: Heads (H) or Tails (T). Since the coin is tossed 3 times, the total number of possible outcomes can be calculated as:

\[ \text{Total Outcomes} = 2^3 = 8 \]

### Step 2: List the possible outcomes
The possible outcomes when tossing the coin three times are:
1. HHH (3 Heads)
2. HHT (2 Heads, 1 Tail)
3. HTH (2 Heads, 1 Tail)
4. THH (2 Heads, 1 Tail)
5. HTT (1 Head, 2 Tails)
6. THT (1 Head, 2 Tails)
7. TTH (1 Head, 2 Tails)
8. TTT (3 Tails)

### Step 3: Determine the winning outcomes for Hanif
Hanif wins if all tosses yield the same result, which can happen in two scenarios:
1. All Heads (HHH)
2. All Tails (TTT)

Thus, there are 2 winning outcomes.

### Step 4: Calculate the losing outcomes
To find the number of losing outcomes, we subtract the winning outcomes from the total outcomes:

\[ \text{Losing Outcomes} = \text{Total Outcomes} - \text{Winning Outcomes} = 8 - 2 = 6 \]

### Step 5: Calculate the probability of losing
The probability of losing is given by the ratio of losing outcomes to total outcomes:

\[ \text{Probability of Losing} = \frac{\text{Losing Outcomes}}{\text{Total Outcomes}} = \frac{6}{8} = \frac{3}{4} \]

### Final Answer
Thus, the probability that Hanif will lose the game is:

\[ \frac{3}{4} \]

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