First slide

Theorems of probability

Question

Two cubes have their faces painted either red or blue. The first cube has five red faces and one blue face. When the two cubes are rolled simultaneously, the probability that the top faces of two cubes shows the same colouris $\frac{1}{2}$ . Then number of red faces on the second cube is

Moderate

A

1
B

2
C

3
D

4

Solution

$Given the number of red faces on the 1^{st} cube is 5$
The number of blue faces on the 1^st cube is 1
Let the number of red faces on the 2^nd cube is x
The number of blue faces on the 2^nd cube is $6 - x$
Probability of getting red face on 1^st cube is $\frac{5}{6}$
Probability of getting blue face on 1^st cube is $\frac{1}{6}$
Probability of getting red face on 2^nd cube is $\frac{x}{6}$
Probability of getting blue face on 2^nd cube is $\frac{6 - x}{6}$
Required probability $P (R R or B B) = \frac{1}{2}$
$\begin{array}{l} \Rightarrow P (RR) + P (BB) = \frac{1}{2} \\ \Rightarrow \frac{5}{6} \cdot \frac{x}{6} + \frac{1}{6} \cdot \frac{6 - x}{6} = \frac{1}{2} \\ \Rightarrow 5 x + 6 - x = 18 \\ \Rightarrow 4 x = 12 \\ \Rightarrow x = 3 \end{array}$
Therefore, the correct answer is (3).

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