# Permutation of lottery?

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In a state of lottery four digits are drawn at random one at a time with replacement from 0 to 9. Suppose that you win if any permutation of your selected integers is drawn. Give the probability of winning if you select

a.6,7,8,9

b.6,7,8,8

c.7,7,8,8

d.7,8,8,8

In a state of lottery four digits are drawn at random one at a time with replacement from 0 to 9. Suppose that you win if any permutation of your selected integers is drawn. Give the probability of winning if you select

a.6,7,8,9

b.6,7,8,8

c.7,7,8,8

d.7,8,8,8

##### 1 Answer

See below:

#### Explanation:

With a permutation, the order of the draw matters. Since we are looking at draws with replacement, each digit has a

probability of our number being drawn.

If, however, the question is saying that with the four drawn numbers they can be rearranged into any permutation, then what we are really talking about is combinations (where order of the draw doesn't matter). These combinations are again done with replacement, and so we need to look at each case separately.

**a**

There is a

**b**

There is a

If we drew an 8 on the first draw (and there is a 50% chance of doing so), then the second, third and fourth draws will be at probabilities of

However, the other 50% of the time we'll draw either the 6 or the 7. If we do so, we then have to look to look a little further for our calculation:

With the second draw (after drawing either a 6 or a 7), we can draw either an 8 (which will happen

If we drew an 8, the third and fourth draws will be at probabilities at

For the third and fourth draws and only 8s remaining, there is a

Let's evaluate:

**c**

There is a

If we drew a 7 (50% chance), then on the second draw if we draw an 8 (

If we drew a 7 on both the first and second (

And evaluate:

**d**

On the first draw, we can only draw a 7 or 8, with a probability of

If we drew a 7 (a

If we drew an 8, we need to look further:

On the second draw (after the first draw of an 8), we can draw either a 7 or 8.

If we drew a 7 (

If we drew an 8, the third and fourth draws will be at

Let's evaluate: