Question

Odds in favour of getting exactly 3 heads in the simultaneous throw of 8 coins in "m to 25" then 'm' is equal to

Answer 1

7

Answer 2

Here's how to solve the problem:

1. Total Possible Outcomes: When you throw 8 coins, each coin has 2 possible outcomes (Heads or Tails). Therefore, the total number of possible outcomes is $2^8 = 256$.

2. Favorable Outcomes: We want to find the number of ways to get exactly 3 heads in 8 throws. This is a combination problem, specifically choosing 3 out of 8 throws to be heads. The number of ways to do this is given by the binomial coefficient $\binom{8}{3}$.

   $$\binom{8}{3} = \frac{8!}{3!(8-3)!} = \frac{8!}{3!5!} = \frac{8 \times 7 \times 6}{3 \times 2 \times 1} = 56$$

   So, there are 56 ways to get exactly 3 heads.

3. Probability of the Event: The probability of getting exactly 3 heads is the number of favorable outcomes divided by the total number of outcomes:

   $$P(\text{exactly 3 heads}) = \frac{\text{Number of ways to get 3 heads}}{\text{Total number of outcomes}} = \frac{56}{256}$$

4. Probability of the Event Not Happening: The probability of not getting exactly 3 heads is $1 - P(\text{exactly 3 heads})$.

   $$P(\text{not exactly 3 heads}) = 1 - \frac{56}{256} = \frac{256 - 56}{256} = \frac{200}{256}$$

5. Odds in Favor: The odds in favor of an event are defined as the ratio of the probability of the event happening to the probability of the event not happening.

   $$\text{Odds in favor} = \frac{P(\text{exactly 3 heads})}{P(\text{not exactly 3 heads})} = \frac{56/256}{200/256} = \frac{56}{200}$$

6. Simplify the Odds: The fraction $\frac{56}{200}$ can be simplified by dividing the numerator and the denominator by their greatest common divisor, which is 8.

   $$\frac{56 \div 8}{200 \div 8} = \frac{7}{25}$$

7. Compare with the Given Odds: The odds in favor are given as "m to 25", which can be written as $\frac{m}{25}$. We calculated the odds to be $\frac{7}{25}$. Comparing the two expressions:

   $$\frac{m}{25} = \frac{7}{25}$$

8. Solve for m: From the equation, it is clear that $m = 7$.