 Hello and welcome to the session. I am Deepika here. Let's discuss the question which says Given two independent events a and b such that probability of a is equal to 0.3 probability of b is equal to 0.6 fine probability of a and b probability of a and not b probability of a or b and probability of neither a nor b now we know that if e and f are independent events then probability of e intersection f is equal to probability of e into probability of f so this is a key idea behind our question we will take the help of this key idea to solve the above question so let's start the solution now according to the question a and b are two independent events such that probability of a is equal to 0.3 and probability of b is equal to 0.6 now in part one we have to find the probability of a and b so in part one we are given probability of a is equal to 0.3 and probability of b is equal to 0.6 now we know that if a and b are two events then the set a intersection b denotes the event a and b so probability of a and b is equal to probability of a intersection b now according to our key idea for independent events a and b probability of a intersection b is equal to probability of a into probability of b now we are given probability of a is equal to 0.3 and probability of b is equal to 0.6 so probability of a and b is equal to 0.3 into 0.6 and this is equal to 0.18 hence the answer for part one is 0.18 now in part two we have to find the probability of a and not b now we know that a minus b is the set of all those elements which are in a but not in b therefore the set a minus b may denote the event a but not b again we know that a minus b is equal to a intersection b complement so again in part two we are given probability of a is equal to 0.3 probability of b is equal to 0.6 now probability of a and not b is equal to probability of a intersection b complement now according to our key idea probability of a intersection b complement is equal to probability of a into probability of b complement now we are given probability of b is equal to 0.6 therefore probability of b complement is equal to 1 minus 0.6 and this is equal to 0.4 hence probability of a and not b is equal to 0.3 into 0.4 and this is equal to 0.12 hence the answer for part two is 0.12 now in part three we have to find the probability of a or b now we know that the union of two sets a and b contains all those elements which are either in a or in b or in both so when the sets a and b are two events associated with the sample space then a union b is the event either a or b or both so a union b is also called a or b so probability of a or b is equal to probability of a union b now probability of a union b is equal to probability of a plus probability of b minus probability of a intersection b now we are given probability of a is equal to 0.3 plus probability of b which is given to us 0.6 minus probability of a intersection b now we have proved that probability of a intersection b is 0.18 so probability of a union b is equal to 0.3 plus 0.6 minus 0.1 hit and this is equal to 0.9 minus 0.1 hit and this is again equal to 0.72 hence the answer for part three is 0.72 now in part four we have to find the probability of neither a nor b so the probability of neither a nor b is equal to probability of a union b complement now we know that probability of a union b complement is equal to 1 minus probability of a union b and this is equal to 1 minus now probability of a union b is equal to 0.72 so this is equal to 0.28 hence the probability of neither a nor b is equal to 0.28 hence the answer for part four is 0.28 so this completes our session i hope the solution is clear to you bye and have a nice day