# trouble undestanding conditional probability: "A group of ten individuals is..."

#### earthman

##### New member
trouble undestanding conditional probability: "A group of ten individuals is..."

I find conditional probability very tricky. Here is an example:

A group of ten individuals is drawing straws from a group of 28 long straws and 2 short straws. If the straws are not replaced, what is the probability, as a percentage, that neither of the first two individuals will draw a short straw?

Ok so... I see that there is a total of 30 straws. The event that we are working with is drawing straws. Since we want neither to draw a short straw, we want them to draw long straws. The first time someone draws a straw is an independent event. They have a 28 out of 30 chance of drawing a long straw. The second time someone draws a straw, if the first straw drawn was long, there is a 27 out of 29 chance of drawing a long straw.

I have figured that much out, but what is the chance that neither of them draw a short straw? Is it as simple as multiplying them together? What part of this has to do with conditional probability? Is it just that the number of total straws goes down by one for the second draw that makes this a conditional probability question?

Thanks

#### Dr.Peterson

##### Elite Member
I find conditional probability very tricky. Here is an example:

A group of ten individuals is drawing straws from a group of 28 long straws and 2 short straws. If the straws are not replaced, what is the probability, as a percentage, that neither of the first two individuals will draw a short straw?

Ok so... I see that there is a total of 30 straws. The event that we are working with is drawing straws. Since we want neither to draw a short straw, we want them to draw long straws. The first time someone draws a straw is an independent event. They have a 28 out of 30 chance of drawing a long straw. The second time someone draws a straw, if the first straw drawn was long, there is a 27 out of 29 chance of drawing a long straw.

I have figured that much out, but what is the chance that neither of them draw a short straw? Is it as simple as multiplying them together? What part of this has to do with conditional probability? Is it just that the number of total straws goes down by one for the second draw that makes this a conditional probability question?

Thanks
Yes, it's that simple.

The probability that both draw long straws is

P(first is long and second is long) = P(first is long) * P(second is long | first was long).

And that's exactly what you did:

P(first is long) = 28/30, and P(second is long | first was long) = 27/29.

#### earthman

##### New member
ok, thanks

Yes, it's that simple.

The probability that both draw long straws is

P(first is long and second is long) = P(first is long) * P(second is long | first was long).

And that's exactly what you did:

P(first is long) = 28/30, and P(second is long | first was long) = 27/29.

Thanks for your help Dr.Peterson. Once I remembered that "|" is the symbol for "such that" your explanation was perfect.

#### Dr.Peterson

##### Elite Member
Thanks for your help Dr.Peterson. Once I remembered that "|" is the symbol for "such that" your explanation was perfect.
Great!