Question

The correct answer is `a=9/2`, but how could `a` be a fraction if George can only be awarded points as whole numbers?

George will play a game at the school fair in which he will toss a penny, a dime, and a quarter at the same time. He will receive 3 points for each coin that lands with heads face up. Let `a` represent the total number of points awarded on any toss of the coins. What is the expected value of `a`?

Answer 1

Because the "expected value" is an average, not a count.

Explanation:

Let's look at all the possibilities with H being heads and T being tails.

`{:("Penny","Dime","Nickel"),(H,H,H),(H,H,T),(H,T,H),(H,T,T),(T,H,H),(T,H,T),(T,T,H),(T,T,T):}`

This table exhausts every conceivable possibility, from tossing three heads, to tossing three tails.

Now, let's add up the points, `3` points for each instance of tossing heads.

`{:("Penny","Dime","Nickel","Points"),(H,H,H,9),(H,H,T,6),(H,T,H,6),(H,T,T,3),(T,H,H,6),(T,H,T,3),(T,T,H,3),(T,T,T,0):}`

The expected value is just the average of the possible points, which is the sum of all the points, divided by the number of trials.

`barp=(9+6+6+3+6+3+3+0)/8=36/8=9/2`

It's like asking, what is the average number of kids the typical American family has. The answer is often 2.5, even though people don't have 0.5 kids!

Answer 2

`"see explanation"`

Explanation:

`"this question has to do with Probability (P)"`

`"the scale of probability is"`

`0<=P<=1`

`"where 0 is impossible and 1 is certain"`

`"if it was certain he obtained a head on each coin then"`

`rArra=(1xx3)+(1xx3)+(1xx3)=9" points"`

`"but the probability of a head "=1/2`

`rArra=(1/2xx3)+(1/2xx3)+(1/2xx3)`

`color(white)(rArra)=3/2+3/2+3/2`

`color(white)(rArra)=9/2`