what is the probability that the sum of the dices is 4 ?

Question

in an experiment, a ball is drawn from a box containing 7 green balls and 10 orange balls. if the ball is green, two dice are rolled. if the ball is orange, three dices are rolled

what is the probability that the sum of the dices is 4?

Answer 1

P(green)*p(4 with 2 dice) + p(orange)*p(4 with 3 dice)

= (7/17 )*(4/36) + (10/17)*(3/216)

Answer 2

There are 17 balls total, so the probability of getting a green ball is 7/17 and the probability of getting an orange ball is 10/17.

If you get a green ball, you roll two dice. There are 36 possible outcomes when rolling two dice (six possibilities on the first die, times six on the other). The possible ways to get a sum of four are 1+3, 2+2, and 3+1, which means there are 3 out of 36 ways to get a sum of four. So the probability of rolling two dice and getting the sum of four is 3/36 = 1/12.

This means that the probability of getting a green ball AND rolling a sum of four is 7/17 * 1/12 = 7/204.

If you get a green ball, you roll three dice. There are 216 possible outcomes when rolling three dice (six times six times six), and the possible ways to get a sum of four are 1+1+2, 1+2+1, and 2+1+1. This means there are 3 out of 216 possible ways to get a sum of four, so the probability when rolling three dice is 3/216 = 1/72.

This means that the probability of getting an orange ball AND rolling a sum of four is 10/17 * 1/72 = 5/612.

Whether you get a green ball or an orange ball, it is still possible to get a sum of 4 after rolling the dice. You can add the probability of getting the 4 with a green ball and getting the 4 with an orange ball to find the total probability of rolling a sum of 4.

7/204 + 5/612 = 21/612 + 5/612 = 26/612 = 13/306.