Quiz - D        

Please simplify all formulas down to powers and the four basic operations.

A 6-sided die and a 20-sided die are rolled together.

(1).- What is the probability that the sum of their faces is equal to 10?

Notice that the dice can be rolled in 6*20 different ways.

So the size of the sample space S is 6*20.

Let E be the event that the sum of the dice is equal to 10.

E = {(1,9), (2,8), (3,7), (4,6), (5,5), (6,4)}

|E| = 6

So for the probability of E :  

p(E) = |E| / |S|

= 6 / 6*20

= 1/20

(2).- What is the probability that the sum of their faces is greater than 22?

Let E be the event that the sum of the dice is greater than 22.

E = {(3,20), (4,20), (5,20), (6,20), (4,19), (5,19), (6,19), (5,18), (6,18), (6,17)}

|E| = 10

So for the probability of E :  

p(E) = |E| / |S|

= 10 / 6*20

= 1/12

-=-

Paul, John, George and Ringo play one match of {rock, paper, scissors }.

Assume that they are equally likely to pick any hand symbol.

(3).- What is the probability that all 4 pick “paper”?

Notice that the friends can pick hand signs in 3^4 different ways.

So the size of the sample space S is 3^4.

Let E be the event that all 4 pick “paper”.

There is only one such outcome, so |E| = 1

So for the probability of E :  

p(E) = |E| / |S|

= 1/ 3^4

(4).- What is the probability that no one picks “scissors”?

Let E be the event that no one picks “scissors”.

There are 2^4 such outcomes, as each friend has two other symbols to choose from, so |E| = 2^4

So for the probability of E :  

p(E) = |E| / |S|

= 2^4 / 3^4

(5).- What is the probability that exactly two of them pick “rock”?  

Let E be the event exactly two of them pick “rock”.

Outcomes of E can be built following a 3 step operation.

Step 1) choose the 2 friends that pick “rock”.

Step 2) pick a non-”rock” sign for a friend not chosen in step 1.

Step 3) pick a non-”rock” sign for the last friend.

Step 1 can be done in (4 choose 2) ways

Step 2 and 3 can be done in 2 ways each.

Therefore:

|E| = (4 choose 2) * 2 * 2

|E| = 4! /2!(4-2)!   * 4

|E| = 4*3*2/2*2    *4

|E| = 3*2  * 4

|E| = 24

So for the probability of E :  

p(E) = |E| / |S|

= 24 / 3^4

-=-

(6).- Fill in the blank:  

p(A | B) = p(A∩B)  / p(B)