Chapter 2, Problem 37RP

To determine

To Find: The amount of money put by Tippy against Bippy to make the bet even.

Answer to Problem 37RP

The amount of money put by Tippy against Bippy for the given condition = $ $14$

Explanation of Solution

Given information:

We are given that the clock cuckoos after every half an hour. Bippy enters the room before Tippy checked clock and made a bet of $ $10$ for the hands of clock forming acute angle. Also, the clock hands have not moved since the cuckoo sounded

Concept Used:

The Probability of an event represents the chance that the event will occur. Probability of even bet means the chances of winning and losing are same. When it is even bet the value of money is obtained by:

Amount won per bet $\times$ probability of winning = Amount lost per bet $\times$ probability of losing

Calculation:

Now here as Bippy has put his money on the formation of acute angle, thus if the angle will not be acute then Tippy will win.

Let Tippy has put $ $x$against Bippy’s $ $10$.

Number of times acute angle is made in $24$ hours:

  $(1:00,2:00,3:30,4:30,5:30)\times 2=5\times 2=10$

Number of times non acute angle is made in $24$ hours: $24-10=14$

Probability of acute angle = $\frac{10}{24}=\frac{5}{12}$

Probability of non-acute angles= $\frac{14}{24}=\frac{7}{12}$

Considering Tippy’s case:

Fixing all these values in the above formula:

Amount won per bet $\times$ probability of winning = Amount lost per bet $\times$ probability of losing

  $$

  $\begin{array}{l}10\times \frac{7}{12}=x\times \frac{5}{12}\\ \\ \Rightarrow x\text{ = 14}\end{array}$

Which will be same if we consider Bippy’s case as it is an even bet.

Conclusion:

The amount of money put by Tippy against Bippy for the given condition = $ $14$