Question

A particular video game has 1 winner for every 3 losers. A second game has 17 winners for every 51 losers. Which game has better odds of winning? If a third game has 18 losers, how many winners are needed to keep the same ratio as the first game?

Answer 1

Hint: We will find the probability of winning in first game and in second game by using the formula of probability \[\dfrac{{{\text{number of favourable outcomes}}}}{{{\text{number of total outcomes}}}}\]. Next compare the probability of both the games to see which game has better odds of winning. We also have to find the number of winners if the number of losers is 18 in the third game. Let the number of winners be $x$ and equate the ratio of winners to losers of the third game to the first game and find the value of $x$.

Complete step-by-step answer:
We are given that in the first game there are 1 loser and 3 winners.
That is, we can say out of 4 people only 1 person wins.
As we know probability of an event is given by \[\dfrac{{{\text{number of favourable outcomes}}}}{{{\text{number of total outcomes}}}}\]
Than we can calculate the chance of winning by dividing 1 by 4
Therefore, chance of winning in first game is $\dfrac{1}{4}$
Similarly in the second game there are 17 winners when there are 51 losers.
That is 17 people will win out of \[17 + 51 = 68\] people
The chances of winning in second game is $\dfrac{{17}}{{68}} = \dfrac{1}{4}$
Thus both the first and second game have the same odds of winning.
Now we want to find the number of winners in the third game if the number of the losers is 18 such that the ratio of winners and losers is the same as that of the first game.
The ratio in the first game of winners to losers is $\dfrac{1}{3}$
Now let the number of winners be $x$
Then $\dfrac{x}{{18}} = \dfrac{1}{3}$
Multiply both sides by 18
$x = 6$
Hence, in the third game there should be 6 winners.

Note: The event with high probability has better odds of winning. But, here both the probabilities are equal. Probability of an event lies between 0 and 1. That is, probability of an event cannot be less than 0 and cannot be greater than 1.