Question

How do you use the counting principle to find the probability of choosing the 5 winning lottery numbers when the numbers are chosen at random from 0 to 9?

Answer 1

Consider the lottery number to be of 5 digits and the repetition of numbers is allowed. Now, choose the first digit and find the probability of choosing it correctly by considering the favourable outcome as 1 and the total number of outcomes as 10. Similarly, find the probability of choosing the other digits correctly and multiply all the probabilities to get the answer.

Complete step by step solution:
Here we have asked to determine the probability of choosing the 5 winning lottery numbers when the digits are from 0 to 9.
Now, 5 winning lottery numbers means the lottery numbers contain 5 digits and we have chosen the correct combination. There will only be one correct combination for the winning lottery.
Let us consider the first digit. So we need to choose the correct digit and that will be one of the digits from 0 to 9. That means the favourable outcome is one and the total number of sample space is 10, therefore we get,
$\Rightarrow$ Probability of choosing the correct digit = $\dfrac{1}{10}$
In the question we haven’t been told that if the digits can repeat or not so we will consider that the digit might repeat. Therefore the probability of choosing the other four digits correctly will also be $\dfrac{1}{10}$ because in those cases also we have to choose the only correct digit from 0 to 9. There will be a total of 5 probabilities for 5 digits. So the overall probability will be the product of all these 5 probabilities because we need to perform all the 5 actions one by one.
$\Rightarrow$ Probability of winning the lottery = $\dfrac{1}{10}\times \dfrac{1}{10}\times \dfrac{1}{10}\times \dfrac{1}{10}\times \dfrac{1}{10}$
$\therefore$ Probability of winning the lottery = $\dfrac{1}{{{10}^{5}}}$
Hence the above obtained value is our answer.

Note: The counting principle is only a different way of representing the combinations and permutations rule. You can think of the situation in a different manner. Consider 5 boxes and think that you have to fill the 5 digits correctly such that you may win the lottery. Since, there will only be a single such combination of 5 digits to the favourable outcome is only 1 which is the correct combination of digits. Now, each box can be filled with 10 digits so the 5 boxes can be filled with ${{10}^{5}}$ combination of digits which is the total number of sample space. Now, you can easily take the ratio to get the answer. In case the digits may not repeat then the total number of sample space will be $10\times 9\times 8\times 7\times 6$.