A bag contains six balls of different colours. Two balls are drawn in succession with replacement. The probability that both the balls are of the same colour is p. Next four balls are drawn in succession with replacement and the probability that exactly three balls are of the same colour is q . If p: q=m: n , where m and n are coprime, then m+n is equal to :

Probability PYQ — JEE Main 2023 · 30 Jan · Shift 2 · Q42 with Solution

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Correct answer: 14

Given situation

There are $6$ balls of different colours, so each draw can result in any one of $6$ colours with equal probability.

Since drawing is with replacement, all draws are independent.

Find $p$

Two balls are drawn in succession with replacement.

We need the probability that both balls are of the same colour.

First ball can be of any colour.
For the second ball to match the first, the probability is $\frac{1}{6}$ .

Hence,

p = \frac{1}{6}

Find $q$

Now $4$ balls are drawn in succession with replacement.

We need the probability that exactly three balls are of the same colour.

This means the colour pattern is:

one colour appears exactly $3$ times,
another different colour appears exactly $1$ time.

Let us count this carefully.

Step 1: Choose the colour that appears 3 times

There are $6$ choices.

Step 2: Choose the colour that appears 1 time

There are $5$ remaining choices.

Step 3: Choose the position of the odd ball

Among the $4$ draws, choose which one is the single different-colour ball:

\binom{4}{1} = 4

So total favourable sequences:

6 \cdot 5 \cdot 4 = 120

Total possible sequences of $4$ draws:

6^4 = 1296

Therefore,

q = \frac{120}{1296} = \frac{5}{54}

Find the ratio $p:q$

p:q = \frac{1}{6} : \frac{5}{54}

Multiply both terms by $54$ :

p:q = 9:5

So,

m = 9, \quad n = 5

Hence,

m+n = 9+5 = 14

Compare with stored answer

Our derived answer is $14$ , which matches the stored correct answer.

More from Probability

View written solutionFree

Correct answer: 14

Given situation

There are $6$ balls of different colours, so each draw can result in any one of $6$ colours with equal probability.

Since drawing is with replacement, all draws are independent.

Find $p$

Two balls are drawn in succession with replacement.

We need the probability that both balls are of the same colour.

First ball can be of any colour.
For the second ball to match the first, the probability is $\frac{1}{6}$ .

Hence,

p = \frac{1}{6}

Find $q$

Now $4$ balls are drawn in succession with replacement.

We need the probability that exactly three balls are of the same colour.

This means the colour pattern is:

one colour appears exactly $3$ times,
another different colour appears exactly $1$ time.

Let us count this carefully.

Step 1: Choose the colour that appears 3 times

There are $6$ choices.

Step 2: Choose the colour that appears 1 time

There are $5$ remaining choices.

Step 3: Choose the position of the odd ball

Among the $4$ draws, choose which one is the single different-colour ball:

\binom{4}{1} = 4

So total favourable sequences:

6 \cdot 5 \cdot 4 = 120

Total possible sequences of $4$ draws:

6^4 = 1296

Therefore,

q = \frac{120}{1296} = \frac{5}{54}

Find the ratio $p:q$

p:q = \frac{1}{6} : \frac{5}{54}

Multiply both terms by $54$ :

p:q = 9:5

So,

m = 9, \quad n = 5

Hence,

m+n = 9+5 = 14

Compare with stored answer

Our derived answer is $14$ , which matches the stored correct answer.

Probability question

Probability question

Step 1: Choose the colour that appears 3 times

Step 2: Choose the colour that appears 1 time

Step 3: Choose the position of the odd ball

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Probability question

Probability question

Step 1: Choose the colour that appears 3 times

Step 2: Choose the colour that appears 1 time

Step 3: Choose the position of the odd ball

More from Probability