Q23:
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.

Solution :

Given: A game involves tossing a fair one-rupee coin 3 times. Hanif wins if all three tosses result in the same outcome (HHH or TTT). Hanif loses if the outcomes are not all the same.

To Find: The probability that Hanif will lose the game.

Step 1: Determining the Sample Space

When a coin is tossed once, there are 2 possible outcomes: Head (H) or Tail (T). When a coin is tossed 3 times, the total number of possible outcomes is $2^3 = 2 \times 2 \times 2 = 8$.

Let $S$ be the sample space representing all possible outcomes:

$S = \{HHH, HHT, HTH, HTT, THH, THT, TTH, TTT\}$

The total number of elementary events, $n(S) = 8$.

Step 2: Identifying Winning Outcomes

Hanif wins if all tosses result in the same outcome. These outcomes are:

Winning outcomes = $\{HHH, TTT\}$

Number of winning outcomes, $n(W) = 2$.

Step 3: Identifying Losing Outcomes

Hanif loses if the outcome is not one of the winning outcomes. These outcomes are:

Losing outcomes = $\{HHT, HTH, HTT, THH, THT, TTH\}$

Number of losing outcomes, $n(L) = n(S) - n(W) = 8 - 2 = 6$.

Step 4: Calculating the Probability of Losing

The probability of an event $E$ is given by the formula:

$P(E) = \frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}}$

Let $L$ be the event that Hanif loses the game.

$P(L) = \frac{n(L)}{n(S)}$

$P(L) = \frac{6}{8}$

Step 5: Simplifying the Fraction

To simplify $\frac{6}{8}$, we divide both the numerator and the denominator by their greatest common divisor, which is 2:

$P(L) = \frac{6 \div 2}{8 \div 2} = \frac{3}{4}$

Alternatively, using the complement rule:

$P(W) = \frac{n(W)}{n(S)} = \frac{2}{8} = \frac{1}{4}$

$P(L) = 1 - P(W)$ [Since the sum of probabilities of complementary events is 1]

$P(L) = 1 - \frac{1}{4} = \frac{3}{4}$

Final Answer: The probability that Hanif will lose the game is $\frac{3}{4}$ or $0.75$.

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Q23: 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.

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Q23:
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.