Question

A has 3 shares in a lottery containing 3 prizes and 9 blanks; B has 2 shares in a lottery containing 2 prizes and 6 blanks: compare their chances of success.

Answer 1

Hint: You are given the shares of A and B already. To solve the question, recollect your knowledge of both probability and combinations. You can check all the combinations of how A will lose on the lottery and then finding the probability of A’s win will be a cakewalk. Repeat the same for B and then compare them to find the final answer.

Complete step-by-step answer:
A’s share in the lottery \[ = 3\]
The lottery has total prizes \[ = 3\]
Total number of blanks in the lottery \[ = 9\]
B’s share in the lottery \[ = 2\]
The lottery has total prizes \[ = 2\]
Total number of blanks in the lottery \[ = 6\]
Step 1:
$\Rightarrow$ Total prizes and blanks when A has participated \[ = 3 + 9 = 12\]
$\Rightarrow$ Total prizes and blanks when B has participated \[ = 2 + 6 = 8\]
Step 2:
$\Rightarrow$ The probability when A can lose = \[\dfrac{{^9{C_3}}}{{^{12}{C_3}}} = \dfrac{{84}}{{220}}\]
$\Rightarrow$ The probability when A can lose = \[\dfrac{{^6{C_2}}}{{^8{C_2}}} = \dfrac{{15}}{{28}}\]
Step 3:
$\Rightarrow$ The probability when A can win = 1 - \[\dfrac{{84}}{{220}}\]\[ = \dfrac{{136}}{{220}}\]
$\Rightarrow$ The probability when B can win = 1 - \[\dfrac{{15}}{{28}}\]\[ = \dfrac{{13}}{{28}}\]
Step 4:
While comparing the ratio of A winning to B winning, we find A: B =\[\dfrac{{136}}{{220}}\]:\[\dfrac{{13}}{{28}}\]
Hence the answer to the question is\[\dfrac{{136}}{{220}}\]:\[\dfrac{{13}}{{28}}\]

Note: To solve such questions, make sure you have a good grasp of permutations, combinations and probability. Probability can never be greater than 1. Such questions can commonly be seen in competitive entrance exams and board exams. Try starting from your basic textbooks to firstly distinguish between permutations and combinations as many students get confused in P&C from the beginning. But with thorough practice it will become easy.