Question

You and a friend are rolling a set of 6 dice. The game works
such that if a die shows the values 1, 2, or 3 you will get a point
for that die. Each die that shows 4, 5, or 6 your friend will get a
point for.

Construct a probability model for a single roll of the dice then
answer the following.

Step 1 of 5:

What is the probability you made 2 points?

Step 2 of 5:

What is the probability that your friend will score 2 points?

Step 3 of 5:

What is the probability you score 4 or more points in this round

Step 4 of 5:

If we play a second round of this game, what is the probability that you will have exactly 6 points at the end of the second round? [In this situation there have been a total of 12 dice rolled]

Step 5 of 5:

If we play another round [a total of 18 dice rolled] what is the probability you will have 10 or more points?

Answer #1

Step 1 of 5: as probability of a point on single die =1/2 (as there are 3 outcome to get a point out of 6)

probability you made 2 points
=^{6}C_{2}(1/2)^{3}(1/2)^{4}
=0.234375

Step 2 of 5: probability that your friend will score 2 points
=^{6}C_{2}(1/2)^{3}(1/2)^{4}
=0.234375

Step 3 of 5: probability you score 4 or more points in this round
=P(X>=4)=P(X=4)+P(X=5)+P(X=6)

=^{6}C_{4}(1/2)^{4}(1/2)^{2}
+^{6}C_{5}(1/2)^{5}(1/2)^{1}
+^{6}C_{6}(1/2)^{6}(1/2)^{0}
=0.343750

Step 4 of 5:
P(X=6)=^{12}C_{6}(1/2)^{6}(1/2)^{6}
=0.225586

Step 5 of 5:

P(X>=10)= =0.407265

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