Below are three boxes containing letters: BOX 1: MATH, BOX 2: AND, BOX 3: HISTORY

Question

(a) From box 1 three letters are drawn one-by-one without replacement and recorded in order. What is the probability that the outcome is HAT?

(b) From box 1 three letters are drawn one-by-one with replacement and recorded in order. What is the probability that the outcome is HAT?

(c) One letter is drawn at random from box 1, then another from box 2, and then another from box 3, with the results recorded in order. What is the probability that the outcome is HAT?

Answer 1

(a)

Let’s look at problem a. Imagine that each of the letters of MATH are placed on to tiles, like in scrabble. Each time we draw from the bag we take the letter and write it down and we leave that letter tile out of the bag. Every time we draw a letter the number of tiles in the bag decrease by 1. In the first draw there are 4 letters, in the second there are 3, and in the third there are 2 letters.

Let’s take a look at our math problem

So let’s say we want to draw the word HAT

when you reach in for your first tile, you have a 1 out of 4 chance of getting an H or a 1/4 chance of getting H.

Now, because we are working on the probability of the word HAT, for finding the probability of the second letter, let us assume we have already drawn an H. Since we cannot put the tile back in there are only 3 tiles left in the bag. There is now a 1 and 3 chance we will get the letter A or 1/3 chance of getting A.

Can you guess what we will do to find T, think now we assume that both H and A are out of the bag now we have 2 tiles left in the bag or a ½ chance of getting T.

To find the total probability of getting HAT multiply all of the fractions together. ¼ x 1/3 x ½ this should equal 1/24

When working this with your student on this, demonstrate this with the word AT and then let them find it using the word HAT.

(b)

This problem is very similar to the last one except every time we draw a letter we put it back into the bag.

So for the word HAT again

When we want to find H there are 4 tiles in the bag, so we have a ¼ chance of getting H

After drawing H we put it back into the bag. We now will go for the letter A, there are 4 tiles in the bag, since we put H back, so again there is a ¼ chance of getting A.

With T it is the same probability as A

So now multiply all of the fractions for each letter together to get the total probability

(c)

This last one is the trickiest. In each box we take the letters of each word and put them on scrabble tiles. Then we get 3 bags, Bag one has letters from box 1, bag 2 from box 2, etc… With this one for each letter we will find the probability of drawing it from each respective bag.

We draw H from bag 1, there are 4 tiles. The probability of getting H is 1/4

For A we draw from bag 2, there are 3 tiles, so the probability is 1/3

For T there are 7 tiles. The probability of T is 1/7

Finally, we multiply the fractions of each letter together to find the probability of getting HAT. It is 1/84.