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HELP for understanding successful outcome value

Screen Link:
https://app.dataquest.io/m/382/guided-project%3A-mobile-app-for-lottery-addiction/7/less-winning-numbers-function

Doubt:
On the screen link shared it says,
For each one of the six five-number combinations above, there are 44 possible successful outcomes in a lottery drawing.
kindly elaborate that how the 44 value is computed as number of successful outcomes?

The content is a bit confusing, but this is what you need to focus on -

  1. You are selecting 6 unique numbers between 1 to 49
  2. We are trying to find the probability that out of those 6, exactly 5 are winning numbers.

Now, when we select 6 numbers, we can have something like - 1, 2, 3, 4, 5, 6

Now, out of these 6 numbers, how many 5 number combinations can we have? We select 5 numbers from the above 6 and can get the following -

1. (1, 2, 3, 4, 5)
2. (1, 2, 3, 4, 6)
3. (1, 2, 3, 5, 6)
4. (1, 2, 4, 5, 6)
5. (1, 3, 4, 5, 6)
6. (2, 3, 4, 5, 6)

That seems straightforward enough. Based on the above, we can see that if we select any 6 numbers, and if we wanted to find out combinations of 5 from those 6, we will always get 6 combinations. That’s what the "6 choose 5" gives us.

Let’s go back to the two points I shared in the beginning -

  1. You are selecting 6 unique numbers between 1 to 49
  2. We are trying to find the probability that out of those 6, exactly 5 are winning numbers.

Looking at the 2nd point, let’s assume that 1, 2, 3, 4, 5 are our 5 winning numbers. We fix that as an example. As per the 1st point, we still need a 6th number, though. Since we have fixed 1 to 5, we can only select from 6 to 49.

That’s what the next part of the content explains us. So, if we fix 1, 2, 3, 4, 5 as our supposed winning numbers, then we can have the following possible list -

  • (1, 2, 3, 4, 5, 6)
  • (1, 2, 3, 4, 5, 7)
  • (1, 2, 3, 4, 5, 8)
  • …
  • (1, 2, 3, 4, 5, 47)
  • (1, 2, 3, 4, 5, 48)
  • (1, 2, 3, 4, 5, 49)

The total number of possible combinations above will be (49 - 6 + 1), that is 44.

We will have 44 possible successful outcomes.

The next part might be even more confusing -

However, we need to leave out the outcome (1, 2, 3, 4, 5, 6) because we’re only interested in outcomes that match exactly five numbers, not at least five numbers. This means that for each of our six five-number combinations we have 43 possible successful outcomes, not 44.

To clarify, if we need to find the probability that exactly 5 are the winning numbers, and we know that the lottery tickets only take numbers from 1 to 49, then that means that out of those 44 lists of numbers above, there can be 1 ticket which might have all 6 numbers as the winning numbers. We don’t want that. We want exactly 5 numbers.

That’s why we get 43 possible successful outcomes.

Hopefully, this clarifies it up a bit. Probability is a tricky subject for me as well.

2 Likes

Thanks a lot, it helped me out to at least understand the concept behind this screen’s content.

1 Like