n(s) = 36 i.e. (1,1)(1,2)(1,3)(1,4)(1,5)(1,6) (2,1)(2,2)(2,3)(2,4)(2,5)(2,6) (3,1)(3,2)(3,3)(3,4)(3,5)(3,6) (4,1)(4,2)(4,3)(4,4)(4,5)(4,6) (5,1)(5,2)(5,3)(5,4)(5,5)(5,6) (6,1)(6,2)(6,3)(6,4)(6,5)(6,6)} Event = {multiple of 3 as a sum}= {(1, 2), (2, 1), (1, 5), (5, 1), (2, 4), (4, 2), (3, 3), (3, 6), (6, 3), (5, 4), (4, 5), (6, 6)}n(E) = 12 p(E) = ? ∴ P(E) = `"n(E)"/"n(S)" = 12/36 = 1/3`. Text Solution Solution : Desired sums of the numbers are 3, 6, 9 and 12,<br>n (S) = 2 + 5 + 4 + 1<brge 12<br> `therefore`Reqd. probability=`12/36=1/3` Answer VerifiedHint: Calculate the number of possible outcomes for throwing two dice. Calculate the number of favourable outcomes for each of the cases. Use the fact that the probability of any event is the ratio of the number of favourable outcomes and the number of possible outcomes to calculate the probability of each of the events. Complete step-by-step solution - We have to calculate the probability of each of the events when two dice are thrown.We know that the total number of possible outcomes when two dice are thrown is $=6\times 6=36$.We know that the probability of any event is the ratio of the number of favourable outcomes and the number of possible outcomes.We will now calculate the probability of events in each case.(a) We have to calculate the probability that the sum of digits is a prime number.We will draw a table showing the sum of digits on rolling both the dice.
When a die is rolled (or thrown), we get the outcomes as: {1, 2, 3, 4, 5, 6} That means, the total number of outcomes = n(S) = 6 Let E be the event of getting a multiple of 3. E = {3, 6} Number of favorable outcomes of the event E = n(E) = 2 P(E) = n(E)/n(S) = 2/6 = ⅓ Hence, the probability of getting a multiple of 3 when a die is thrown is 1/3.
$\begingroup$
What is the probability that the sum of $2$ dice rolls is a multiple of $3$? What about for $3$ dice rolls? For $n$ dice rolls? So I have the first part of this solution worked out by writing out all the combinations of $2$ dice rolls (I won't write them here). There are $6^2=36$ total and the following multiples of $3$ are rolled: $$3\rightarrow2 ways$$$$6\rightarrow5ways$$$$9\rightarrow4ways$$$$12\rightarrow1way$$So adding up all the ways over the total rolls is the probability:$$\frac{2+5+4+1}{6^2}=\frac{12}{36}=\frac{1}{3}=0.33$$The second part and the general solution is what trips me up. I know there are $6^3$ possible rolls for $3$ dice, and we're now including the multiples $15$ and $18$, but how do I figure out how many ways there are to roll each multiple without counting all $216$ possibilities? How would I apply this to a general solution with $n$ dice?
$\endgroup$ |