1. ## Combinations in probability

1. Joseph throws two dice, one blue and the other yellow, both numbered from 1 to 6. The probability that the blue die rolls a higher number than the yellow die is?

2. 10 cards numbered 1 to 10 are placed in a box. First, Joseph picks a card randomly and then Mary picks one, also randomly. What is the probability that the sum of Joseph and Mary's cards is an even number?

2. $\begin{gathered}
(2,1) \hfill \\
(3,2),(3,1) \hfill \\
(4,3),(4,2),(4,1) \hfill \\
(5,4),(5,3),(5,2),(5,1) \hfill \\
(6,5),(6,4),(6,3),(6,2),(6,1) \hfill \\
\end{gathered}$

3. 2. There are ${10\choose 2}=45$ equally likely pairs, ignoring order and who picked what.
We need to figure out which of these pairs are even and which are odd.
At first I thought the answer should be .5. But we have 45 sample points.
List them as above
(1,2).....(1,10)
(2,3).....(2,10)
.
.
(9,10)
and figure how many are even and odd.
Take the number of even ones divided by 45 should be the answer.
I counted across and think there are 20 even favorable simple events.
Look it over, if that's correct, the answer is 20/45=4/9.

4. ## Probability

Hello HFBA
Originally Posted by HFBA
1. Joseph throws two dice, one blue and the other yellow, both numbered from 1 to 6. The probability that the blue die rolls a higher number than the yellow die is?
An alternative to listing all the possible pairs is to say that out of the 36 possible ways in which two dice can land, 6 show equal scores. In half of the remaining 30 ways, the blue die's score will be higher than the yellow. That's 15 out of 36 altogether, giving a probability of $\frac{15}{36}= \frac{5}{12}$.