If you are rolling 3 dice, what is the probability of rolling 2 sixes?

Question

Julia Adams · Accepted Answer

Probability of rolling two sixes in 3 dice is #15/216# This is an example of binonmial distribution of general form #X~B(n,p)# Here #n=3# i.e no of dice thrown. and #p=1/6#i.e probability of rolling #6# in single dice. #P(n,r(2)) = nc_r p^r*q^(n-r)# where #r=2# (rolling of two sixes) and #q=5/6# i.e probability of not rolling #6# in single dice. So probability of rolling two sixes in 3 dice is #P(n,r(2)) = nc_r p^r*q^(n-r) or 3c_2*(1/6)^2.(5/6)^(3-2) =3*1/36*5/6=15/216#[Ans]

Hazel Anglin · Answer

undefined To find the probability of rolling 2 sixes with 3 dice, you can use the combination formula:

Number of favorable outcomes / Total number of outcomes

Number of favorable outcomes:
- There are 3 dice rolls, and you want exactly 2 sixes. The third die can be any number other than six. So, you have C(3,2) * 1 * 5 = 3 * 5 = 15 favorable outcomes.

Total number of outcomes:
- Each die has 6 possible outcomes, so for 3 dice, there are 6^3 = 216 total outcomes.

Probability = Number of favorable outcomes / Total number of outcomes
Probability = 15 / 216
Probability ≈ 0.0694 or about 6.94%.

HIX Tutor · Answer

undefined Great question! Let’s figure this out step by step.

1. **What is the total number of outcomes when rolling 3 dice?**
   - Each die has 6 faces. So, if you roll 3 dice, the total outcomes are:
   - 6 (for the first die) × 6 (for the second die) × 6 (for the third die) = 6^3 = 216.

2. **What does "rolling 2 sixes" mean?**
   - It means 2 dice show a 6 and 1 die shows something else.

3. **How many ways can we choose which dice show a 6?**
   - You can choose 2 out of the 3 dice to show 6. This can be done in:
   - $\binom{3}{2} = 3$ ways.

4. **What can the third die be?**
   - The third die can be any number except 6 (1, 2, 3, 4, or 5). So there are 5 options.

5. **Now, let’s calculate the favorable outcomes.**
   - We have 3 ways to choose the dice showing 6, and for each of those, we have 5 choices for the third die:
   - Total favorable outcomes = 3 (ways to choose dice) × 5 (choices for the third die) = 15.

6. **What is the probability of rolling 2 sixes?**
   - Probability = (Favorable outcomes) / (Total outcomes).
   - Probability = 15 / 216.

7. **Can we simplify it?**
   - Yes! Both numbers can be divided by 3.
   - So, 15 ÷ 3 = 5 and 216 ÷ 3 = 72.
   - The simplified probability is: 5 / 72.

So, what do you think is the final answer for the probability of rolling 2 sixes?