NIMCET 2008 — Mathematics PYQ

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Question

NIMCET 2008 — Mathematics PYQ

NIMCET | Mathematics | 2008

A bag contains 6 red and 4 green balls. A fair dice is rolled and a number of balls equal to that appearing on the dice is chosen from the urn at random. The probability that all the balls selected are red is:

Choose the correct answer:

Explanation

1. Analyze the Components

Total balls in bag: $6 (\text{Red}) + 4 (\text{Green}) = 10 \text{ balls}$ .
Dice Outcome ( $i$ ): A fair die has outcomes $i \in \{1, 2, 3, 4, 5, 6\}$ .
Probability of each die face: $P(D=i) = 1/6$ for all $i$ .

2. Condition for "All Red Balls"

Let $E$ be the event that all selected balls are red.

The number of balls selected ( $i$ ) depends on the die. Since there are only 6 red balls available, it is possible to select up to 6 red balls.

The probability of selecting $i$ red balls given that the die shows $i$ is:

P(E|D=i) = \frac{\binom{6}{i}}{\binom{10}{i}}

3. Apply the Law of Total Probability

The total probability $P(E)$ is the sum of probabilities for each possible die roll:

P(E) = \sum_{i=1}^{6} P(D=i) \times P(E|D=i)

P(E) = \frac{1}{6} \left[ \frac{\binom{6}{1}}{\binom{10}{1}} + \frac{\binom{6}{2}}{\binom{10}{2}} + \frac{\binom{6}{3}}{\binom{10}{3}} + \frac{\binom{6}{4}}{\binom{10}{4}} + \frac{\binom{6}{5}}{\binom{10}{5}} + \frac{\binom{6}{6}}{\binom{10}{6}} \right]

4. Calculate each term

$i=1: \frac{6}{10}$
$i=2: \frac{15}{45} = \frac{1}{3}$
$i=3: \frac{20}{120} = \frac{1}{6}$
$i=4: \frac{15}{210} = \frac{1}{14}$
$i=5: \frac{6}{252} = \frac{1}{42}$
$i=6: \frac{1}{210}$

5. Summing the probabilities

P(E) = \frac{1}{6} \left[ \frac{6}{10} + \frac{1}{3} + \frac{1}{6} + \frac{1}{14} + \frac{1}{42} + \frac{1}{210} \right]

To add these, find a common denominator (210):

P(E) = \frac{1}{6} \left[ \frac{126 + 70 + 35 + 15 + 5 + 1}{210} \right]

P(E) = \frac{1}{6} \left[ \frac{252}{210} \right]

P(E) = \frac{1}{6} \times \frac{6}{5} = \frac{1}{5}

Conclusion

The calculated probability is $1/5$ (or $0.2$ ). Since $1/5$ is not listed in options (a), (b), or (c):

Correct Option: (d) None