Q11:
Gopi buys a fish from a shop for his aquarium. The shopkeeper takes out one fish at random from a tank containing 5 male fish and 8 female fish (see Fig. 14.4). What is the probability that the fish taken out is a male fish?

Solution :

Given:

Number of male fish in the tank ($n_m$) = $5$

Number of female fish in the tank ($n_f$) = $8$

To find:

The probability ($P$) that the fish taken out at random is a male fish.

Step 1: Calculate the total number of possible outcomes.

The total number of fish in the tank represents the total number of equally likely outcomes ($N$).

$N = n_m + n_f$

$N = 5 + 8$

$N = 13$

[Since the shopkeeper takes out one fish at random, any of the 13 fish has an equal chance of being selected.]

Step 2: Identify the number of favorable outcomes.

Let $E$ be the event of taking out a male fish.

The number of favorable outcomes for event $E$ is the number of male fish in the tank.

$n(E) = 5$

Step 3: Apply the probability formula.

The probability of an event $E$ is defined as the ratio of the number of favorable outcomes to the total number of possible outcomes.

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

$P(E) = \frac{n(E)}{N}$

Step 4: Perform the calculation.

$P(E) = \frac{5}{13}$

[Since 5 and 13 are coprime numbers, the fraction is in its simplest form.]

Final Answer: The probability that the fish taken out is a male fish is $\frac{5}{13}$.

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