 Hello and welcome to the session that has discussed the following question. It says check whether the following probabilities probability a and probability of b are consistently defined. In the first part we are given probability of a is 0.5 probability of b is 0.7 and probability of a intersection b is 0.6. And in the second part we are given probability of a is 0.5 probability of b is 0.4 and probability of a union b is 0.8. Let us first understand the key idea behind this question. The probability of a union b is equal to probability of a plus probability of b minus probability of a intersection b. So this implies probability of a intersection b is equal to probability of a plus probability of b minus probability of a union b. Now to check the consistency of the probabilities that is probability a and probability of b are consistent probability of a intersection b should be less than equal to probability of a and probability of a intersection b should be less than equal to probability of b. So this knowledge will work as key idea. Let us now move on to the solution. In the first part we are given probability of a is 0.5 probability of b is 0.7 and probability of a intersection b is 0.6. And we can see that the probability of a intersection b is greater than probability of a that is 0.6 is greater than 0.5. So these probabilities are not consistent because for the probabilities to be consistent probability of a intersection b should be less than equal to probability of a and probability of a intersection b should also be less than equal to probability of b. So not consistent. Now in the second part we are given that probability of a is 0.5 probability of b is 0.4 and probability of a union b is 0.8. Now from this we can find out the probability of a intersection b probability of a intersection b is probability of a plus probability of b minus probability of a union b. The probability of a is 0.5 probability of b is 0.4 probability of a union b is 0.8. So probability of a intersection b is 0.1 so probability of a intersection b is less than probability of a that is 0.5 is less than probability of a intersection b is 0.1. The 0.1 is less than probability of a which is 0.5 also probability of a intersection b is less than probability of b probability of a intersection b is 0.1 and probability of b is 0.4. So the probabilities are consistent hence probability is given in the first part of the question are not consistent. So the answer is no because probability of a intersection b must be less than or equal to probability of a and b and the probability is given in the second part of the question are consistent. So the answer is yes and this completes the question. Bye for now. Take care. Have a good day.