Probability Question

Question

A bag contains 4 balls that are numbered 1,2,3,4. Two balls are drawn from the bag without replacement. Find the probability that the number sum to 5.

Answer 1

This is a "conditional" or "interdependent" probability question because the probability of choosing the appropriate ball the second time is dependent on what ball is picked the first time. So, you multiply those two probabilities.

 

The combinations of numbers that sum to 5 are 4+1 and 3+2. You would need to pick in those pairs, but the probabilities of each pair don't change because neither par share a value. From these pairs, the probability of picking of of these numbers is 100% because all 4 are in the group:

 

4/4

 

The probability of then, without replacement (i.e., you only have 3 to choose from), of the number matching one of the pairs above is 1 out of 3, or 1/3.

 

If you first choose a 4, you need a 1 and that probability is 1 out of 3. If you pick a 2, the probability of then picking a 3 is 1 out of 3.

 

So: 4/4 × 1/3 = 4/12 or 1/3. Hope this helps.

Answer 2

Hi Cassie!

 

As on your other probability question...your probability is always going to be "the number of things we care about" divided by "the total number of things there are".

 

In this case, we need to find the number of draws we care about and the total number of draws there are.  Let's do them each in turn:

 

The draws we care about

There are only a few ways to draw balls from the bag without replacement that sum to five.  We could draw 1 and then 4 or 4 and then 1 or 2 and then 3 or 3 and then 2.  That's a total of 4 draws that give us a sum of 5.

 

The total number of draws

There are four ways to draw the first ball because we can get any number 1-4.  On the second draw, since we are not replacing the first ball, we will have 3 ways to draw that one.  (The four balls 1-4, minus the one we already drew on the first draw.)  So if we have 4 choices for the first one and 3 choices for the second one, then we have 4*3 ways to draw balls.  4*3=12.

 

So our probability of drawing a sum of 5 is the draws we care about (4) divided by the total number of draws there are (12):

 

4/12

 

Hope that helps.  Let me know if you have any follow up questions.

 

Jeff