Question

it is presentation day in class and your instructor is drawing names from a hat to determine the order of the presentations. If there are 19 students in the class, what is the probability that the first 3 presentations will be by Mika, Todd and Kelly? Express your answer as a fraction.

Answer 1

The probability of an event is the number of outcomes in which the event happens divided by the number of total outcomes.

The outcome we want is that the first 3 will be Mika, Todd and Kelly, in this order.

For this, we can thing in picks:

The first pick we can only pick Mika, so there is 1 possibilitie1:

`1`

For the second pick, we can pick only Todd, so 1 possibility:

`1*1`

And for the third there is only Kelly, so 1 possibility:

`1*1*1`

For the following pick, we can pick any of the 16 students left, so 16 possibilities:

`1*1*1*16`

And for the next we have one less, so 15 and so on until we picked everyone:

`1*1*1*16*15*14*13*12*11*10*9*8*7*6*5*4*3*2*1`

We don't need to multiply all of this, because some of them will cancel afterwards.

Let's go the all possible outcomes. In this case, the first pick can be any of the 19 students:

`19`

And the next we have one less, and so on:

`19*18*17*16*15*14*13*12*11*10*9*8*7*6*5*4*3*2*1`

The probability will be the number of outcomes of the event we want over the number of total outcomes, so:

`P=(1*1*1*16*15*14*13*12*11*10*9*8*7*6*5*4*3*2*1)/(19*18*17*16*15*14*13*12*11*10*9*8*7*6*5*4*3*2*1)`

Notice that all from 16 down will cancel out, so:

`P=(1*1*1)/(19*18*17)=(1)/(5814)`

So, the probability is 1/5814.