# If two dice are thrown $5$ times, what is the probability of obtaining the same number on both dice $2$ times?

How do I go about solving this question on probability?

Two dice are thrown 5 times. Calculate the probability to obtain the same number on both dice 2 times.

One thing i know that to obtain any number on one die is 1/6. Need help in moving ahead.

• Can you solve: "Two dice are thrown 1 time. Calculate the probability to obtain the same number on both dice"?
– Did
Apr 16, 2017 at 9:08
• Then, can you solve: "The same experiment, with probability of success p, is performed 5 times. Calculate the probability that it is successful 2 times"?
– Did
Apr 16, 2017 at 9:10

$\frac 6 {36}$ of these rolls have the dice the same number, $\{11,22,33,44,55,66\}$, and $\frac {30} {36}$ don't have the same number.
Now think about tossing two dice 5 times. How can you get matching dice exactly two of the five tosses? First and second toss of the two dice or first and third toss or...or fourth and fifth toss. There are $\binom 5 2=10$ ways to choose two of five tosses. The rest of the tosses (three of them) will have non-matching dice. Each of the two tosses with matching dice will give a probability of $\frac 6 {36}$. Each of the three tosses with different dice will give $\frac {30} {36}$.
We end up with our probability. $\binom 5 2(\frac 6 {36})^2(\frac {30} {36})^3$