JEE Mathematics 2023 Question with Solution

Given: The die has faces $-2, -1, 0, 1, 2, 3$ and it is thrown $5$ times.

Find: The probability that the product of the five outcomes is positive.

From the solution, the product is counted as positive when the number of negative outcomes is even and no outcome is $0$.

There are $3$ positive faces and $2$ negative faces, so

$$p = \frac{3}{6} = \frac{1}{2}, \qquad q = \frac{2}{6} = \frac{1}{3}$$

The approach shown on the page adds the cases with an even number of negative outcomes:

$$P(\text{positive}) = {}^5C_5\left(\frac{1}{2}\right)^5 + {}^5C_2\left(\frac{1}{3}\right)^2\left(\frac{1}{2}\right)^3 + {}^5C_4\left(\frac{1}{3}\right)^4\left(\frac{1}{2}\right)$$

Now evaluate each term:

$${}^5C_5\left(\frac{1}{2}\right)^5 = \frac{1}{32}$$ $${}^5C_2\left(\frac{1}{3}\right)^2\left(\frac{1}{2}\right)^3 = 10 \cdot \frac{1}{9} \cdot \frac{1}{8} = \frac{5}{36}$$ $${}^5C_4\left(\frac{1}{3}\right)^4\left(\frac{1}{2}\right) = 5 \cdot \frac{1}{81} \cdot \frac{1}{2} = \frac{5}{162}$$

So,

$$P(\text{positive}) = \frac{1}{32} + \frac{5}{36} + \frac{5}{162} = \frac{243 + 1080 + 240}{7776} = \frac{1563}{7776} = \frac{521}{2592}$$

Therefore, the probability is $\frac{521}{2592}$. The solution explicitly marks the correct option as D, even though this value appears as option B in the listed options. the answer is recorded as D.

Given: Positive faces are $1,2,3$ and negative faces are $-1,-2$. The face $0$ makes the product zero, so it is excluded in favorable cases.

Find: Probability that the product is positive after $5$ throws.

The product is positive when the number of negative outcomes is even. For $5$ throws, the possible even counts are $0,2,4$.

1. No negative outcomes

$${}^5C_5\left(\frac{1}{2}\right)^5 = \frac{1}{32}$$

2. Exactly two negative outcomes

$${}^5C_2\left(\frac{1}{3}\right)^2\left(\frac{1}{2}\right)^3 = \frac{5}{36}$$

3. Exactly four negative outcomes

$${}^5C_4\left(\frac{1}{3}\right)^4\left(\frac{1}{2}\right) = \frac{5}{162}$$

Adding,

$$\frac{1}{32} + \frac{5}{36} + \frac{5}{162} = \frac{521}{2592}$$

Hence the required probability is $\frac{521}{2592}$.