How many of the possible outcomes contain at least one red tile? Full question below

Question

A jar contains only one blue tile, one green tile, one red tile, and one yellow tile. If a tile is chosen at random and then replaced in the jar, and then a second tile is chosen at random, how many of the possible outcomes contain at least one red tile?

 

I know the "answer".  What I would like to know is the equation that gets the number of red outcomes.   I know the total number of outcomes is from 4x4=16 and then I know logically and/or by chart that 7 of those combinations have a red in them.  What I'd like is how to calculate  rather than count to get the 7.   In other words what would you do if the number of outcomes was too great to chart and count?

Answer 1

Hi Judith, I've read your comments and thought I would offer a different view point. Although the problem is simple, we know nothing is ever simple in math as we move forward. Sometimes it is easier to understand the more complex way that could be applied to other scenarios.

* You were able to identify that you needed to use the counting principle of 4 x 4 =16 in order to identify the total number of outcomes, so I assume you caught the first key word "replace". Because the 1st tile was replaced before the 2nd tile was chosen at random, then you know both the first and the second tile chosen had the same chance/ probability of being chosen. 1/4 (because the total sample set is 4 and each color only occurs 1 time.)

 

* You were able to identify the total number of possible outcomes is 16. So basically, you are looking at a pair of numbers that will be pulled from a set of 4 and another duplicate set of 4 (you know the original set of 4 is repeated 1 time. This is far more complicated than it needs to be, but to help you understand the probability of any event occurring is expressed by the total number of the event/ total number of the sample. In this case the first tile chosen would be 1 event/4 samples= 1/4. Logically, you can look at the word problem and know that there are only 4 tiles. each are different color, so the probability of getting any 1 specific color would be 1/4.

 

*When the original tile is replaced and a second tile is chosen as part of the pair, that's where your counting principle takes place to determine total number of outcomes=16.

 

*First take the probability of randomly selecting any 1 color. 1/4 or 2/8 which = 1/4 (because there are only 4 different colors) multiplied by the total number of possible outcomes= 1/4 x 16/1, cross multiply and you get 4.

 

*But now you have to add the number of possibilities of getting at least 1 red out of the pair of 2 tiles that are chosen at random with equal opportunities. In this case, the sample size has shrunk down to 1 (red) and the event is the total number of possibilities of pairs including red. {0,1,2} So your pair could include either 0 reds, 1 reds, or 2 reds.

 

*Again, take the event size/ sample size= 3/1= 3

 

*4 + 3=7

Far more complicated than needed, but sometimes it helps put everything into perspective.

The key words were replace, fact there are only 4 varieties of color. Hope this wasn't too confusing.

 

Kellie

Answer 2

Hi Judith,

 

You could figure out that  there are 4*4=16 total possible outcomes (as you did

 

You could then figure out how many ways you could pick 2 that would not have a red tile 3*3 =9

 

So 9 out of 16 would not have a red tile

 

that means that 16-9 = 7  (would have a red tile

 

Hope this helps