In a probability experiment, two fair six-sided dice were rolled 180 times, yielding an ex… — JAMB UTME

In a probability experiment, two fair six-sided dice were rolled $180$ times, yielding an experimental probability of $\frac{5}{18}$ for obtaining a sum divisible by $3$ . If $m$ additional consecutive rolls were conducted and every single one resulted in a sum divisible by $3$ , the updated overall experimental probability equaled the theoretical probability that the absolute difference between the numbers shown on two fair six-sided dice is at most $1$ . Calculate the value of $m$ .

Answer: 54

Answer

The initial number of successful trials is

180 \times \frac{5}{18} = 50

. The theoretical probability of rolling two dice with an absolute difference of at most

1

is calculated by counting

6

outcomes with difference

0

and

10

outcomes with difference

1

, giving

\frac{16}{36} = \frac{4}{9}

. Equating the updated experimental probability

\frac{50 + m}{180 + m}

\frac{4}{9}

yields

9(50 + m) = 4(180 + m)

, which simplifies to

5m = 270

m = 54

Step-by-Step Solution

Calculate the initial number of successful trials from the given experimental probability.

Initial successful outcomes =

180 \times \frac{5}{18} = 50

Experimental probability is defined as the number of successful trials divided by the total number of trials.

Calculate the theoretical probability that the absolute difference between two rolled six-sided dice is at most 1.

Favorable outcomes = 16, so

P(\text{theoretical}) = \frac{16}{36} = \frac{4}{9}

Outcomes with difference 0:

(1,1), (2,2), (3,3), (4,4), (5,5), (6,6)

(6 outcomes). Outcomes with difference 1:

(1,2), (2,1), (2,3), (3,2), (3,4), (4,3), (4,5), (5,4), (5,6), (6,5)

(10 outcomes). Total sample space

= 6 \times 6 = 36

Formulate the algebraic equation relating the updated experimental probability to the theoretical probability.

\frac{50 + m}{180 + m} = \frac{4}{9}

Adding

m

consecutive successful rolls increases both the number of successful outcomes (to

50 + m

) and the total number of trials (to

180 + m

Solve the equation for

m

9(50 + m) = 4(180 + m) \implies 450 + 9m = 720 + 4m \implies 5m = 270 \implies m = 54

Cross-multiplication converts the rational expression into a linear equation.

Key Concept

Experimental and Theoretical Probability Synthesis

Estimated Time:3m 0s

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