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Question: Two unbiased dice are thrown simultaneously. What is the probability of getting at most one five in ...

Two unbiased dice are thrown simultaneously. What is the probability of getting at most one five in a single throw of the two dice?

Explanation

Solution

Here we have been given that two dice are thrown simultaneously and we have to find the probability of getting at most five in a single throw of the two dice which means the sum of numbers in two dice should be 55 or less than five. Using this we will form our possible outcome and divide it by the total outcome to get the desired answer.

Complete step by step answer:
We have been given that two dice are thrown simultaneously.
Total number of outcome from one dice =6=6
Total number of outcome from two dice =6×6=6\times 6
So number of combination when two dice are rolled is,
n(S)=36n\left( S \right)=36….(1)\left( 1 \right)
Where S=S=Sample Space

Now as we what the sum on the two dice at most 55 so we can have the sum as 2,3,4,52,3,4,5.
Number of possible combination will be,
Sum is 22 - (1,1)\left( 1,1 \right)
Sum is 33 - (1,2),(2,1)\left( 1,2 \right),\left( 2,1 \right)
Sum is 44 - (1,3),(2,2),(3,1)\left( 1,3 \right),\left( 2,2 \right),\left( 3,1 \right)
Sum is 55 - (1,4),(2,3),(3,2),(4,1)\left( 1,4 \right),\left( 2,3 \right),\left( 3,2 \right),\left( 4,1 \right)

Total number of combination is 1+2+3+41+2+3+4.
n(E)=1+2+3+4n\left( E \right)=1+2+3+4
⇒n(E)=10\Rightarrow n\left( E \right)=10….(2)\left( 2 \right)
Where E=E= Event of getting sum at most 55
Probability of getting the sum at most 5=n(E)n(S)5=\dfrac{n\left( E \right)}{n\left( S \right)}
Substitute the value from equation (1) and (2) above we get,
Probability of getting the sum at most 5=10365=\dfrac{10}{36}
Probability of getting the sum at most 5=5185=\dfrac{5}{18}

Hence probability of getting at most one five in a single throw of the two dice is 518\dfrac{5}{18}.

Note: Don’t get confused with at most and at least as at most means less than and equal to and at least mean greater than and equal to. Always write down the possible way we can get the given outcome as it becomes easy to calculate that way. Probability is the possibility of an event occurring or how likely an event can occur under the given condition.