In one urn there are 5 white and 7 black balls in the other 2 white and 5 black balls.

In one urn there are 5 white and 7 black balls in the other 2 white and 5 black balls. What is the probability that 2 black balls will be pulled out of each urn

The total number of outcomes when choosing 2 balls out of 12:
C (12,2) = 12! / (2! (12 – 2)!) = 11 12 / (1 2) = 66;
Number of ways to choose 2 black balls out of 7:
C (7,2) = 7! / (2! (7 – 2)!) = 6 7 / (1 2) = 21;
The number of ways not to take out a single white ball out of 5:
C (5.0) = 1;
Probability of pulling 2 black balls from the first urn:
P2 = C (7.2) C (5.0) / C (12.2) = 21 1/66 = 0.318.
The second urn contains 2 + 5 = 7 balls.
The total number of outcomes when choosing 2 balls out of 7:
C (7,2) = 7! / (2! (7 – 2)!) = 6 7 / (1 2) = 21;
Number of ways to choose 2 black balls out of 5:
C (5,2) = 5! / (2! (7 – 2)!) = 4 5 / (1 2) = 10;
The number of ways not to take out a single white ball out of 2:
C (2.0) = 1;
Probability of pulling 2 black balls from the second urn:
P1 = C (5.2) C (2.0) / C (7.2) = 10 1/21 = 0.476.
These are independent events, the probability of their occurrence together is found by the multiplication theorem:
P = P1 P2 = 0.318 0.476 = 0.151.
Answer: 0.151.