Question

A square dartboard has sections numbered from 1 to 12 as shown below. Players have to make a prediction and throw a dart. They win if their dart lands on the section that matches their prediction. Three friends Amit, Aryan and Aman took the challenge.

Based on the above information, answer the following:

1. What is the probability of Amit winning if he predicted that his dart will land on a prime number?
2. What is the probability of Aryan winning if he predicted that his dart will land on an odd number?
3. What is the probability of Aman winning if he predicted that his dart will land on a square number?
4. Which of the three friends has the highest chance of winning?

Answer 1

The dartboard has 12 equally likely sections numbered 1 to 12, so the total number of outcomes is 12.

By formula,

Probability of an event = \[\frac{\text{Number of favourable outcomes}}{\text{Total number of outcomes}}\]

i. Prime numbers from 1 to 12 are 2, 3, 5, 7 and 11, i.e. 5 numbers.

∴ P(Amit wins) = \[\frac{5}{12}\]

Hence, the probability of Amit winning = \[\frac{5}{12}\]

ii. Odd numbers from 1 to 12 are 1, 3, 5, 7, 9 and 11, i.e. 6 numbers.

∴ P(Aryan wins) = \[\frac{6}{12} = \frac{1}{2}\]

Hence, the probability of Aryan winning = \[\frac{1}{2}\]

iii. Square numbers from 1 to 12 are 1, 4 and 9, i.e. 3 numbers.

∴ P(Aman wins) = \[\frac{3}{12} = \frac{1}{4}\]

Hence, the probability of Aman winning = \[\frac{1}{4}\]

iv. Comparing the three probabilities :

\[\frac{5}{12} \approx 0.42, \quad \frac{1}{2} = 0.5, \quad \frac{1}{4} = 0.25\]

Since \[\frac{1}{2}\] is the largest, Aryan has the highest chance of winning.

Hence, Aryan has the highest chance of winning.