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# Probability and Combinations

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Practice Probability and Combinations
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## Real World Applications of CombinationProblems

### Topic

Chance in Lottery Games

• Combination
• Permutation
• Probability

### Student Exploration

#### What are your chances of winning the lottery?

The California Fantasy 5 game is a state lottery game played across the state. Anyone eighteen-years-old or older can buy a ticket at a local convenience store. All you have to do is pick five lucky numbers from 1 to 39 on a Fantasy 5 slip. Then every day at 6:30pm entries are closed and the state draws the five Fantasy 5 numbers for the day. Now for the winnings! You win if you get one of the four following outcomes. You pick 2 of the 5 winning numbers, you pick 3 of the 5 winning numbers, you pick 4 of the 5 winning numbers, and you pick 5 of the 5 winning numbers.
What is the probability of picking 5 of the 5 winning numbers?
$\#$ of ways to pick the 5 numbers from the 39 is: $_{39}C_5 = \frac{39!}{34! \ 5!}= 575,757$
Therefore the probability of picking 5 of the 5 numbers is $\frac{1}{575,757}$.

### Extension Investigation

a. Why does this situation represent a Combination?
It is a Combination because the order that the numbers are picked doesn’t matter, it is only important that you picked the correct numbers. If it were to be a Permutation then you would also need to also pick the numbers in the correct order, in addition to picking the correct numbers.
b. How many different ways can all five of the Fantasy 5 numbers be picked?
$\#$ of ways to pick the 5 numbers from the 39 is: $_{39}C_5 = \frac{39!}{34! \ 5!}= 575,757$
c. How many different ways can four of the five Fantasy 5 numbers be picked?
$\#$ of ways to pick the 4 numbers from the 39 is: $_{39}C_4 =\frac{39!}{35! \ 4!}= 82,251$
d. If you were analyzing how many ways that two of the five numbers could be picked, how would you approach solving this problem differently?
For this problem you would set up the problem as $r=2$ instead of $r = 4 \ or \ 5$.
e. If you were asked to find the probability of picking four of the five Fantasy 5 numbers correctly, how would you approach that task?
You would need to find the number of ways to pick four numbers from the 39 numbers and then set that number up as the denominator and one as the numerator, as there is one way to correctly pick 4 of the Fantasy 5 numbers.
f. What would need to change in the problem in order for it to represent a Permutation?
If it were to be a Permutation, then you would also need to also pick the numbers in the correct order, in addition to picking the correct numbers.

### Connections to other CK-12 Subject Areas

• Permutations Problems
• Permutations and Combinations Compared
• Theoretical and Experimental Probability