Select correct alternative from the given choices. If two events A and B are such that P(A)=0.4,

Select correct alternative from the given choices. If two...

Select correct alternative from the given choices. If two events $\mathrm{A}$ and $\mathrm{B}$ are such that $\mathrm{P}(\overline{\mathrm{A}})=0.4, \mathrm{P}(\mathrm{B})=$ $0.7$ and $\mathrm{P}(\mathrm{A} \cap \mathrm{B})=0.2$, then $\mathrm{P}\left(\frac{\mathrm{B}}{\mathrm{A} \cup \overline{\mathrm{B}}}\right)$ is (A) $\frac{3}{5}$ (B) $\frac{2}{5}$ (C) $\frac{1}{4}$ (D) $\frac{4}{5}$

Select correct alternative from the given choices.
If two events $\mathrm{A}$ and $\mathrm{B}$ are such that $\mathrm{P}(\overline{\mathrm{A}})=0.4, \mathrm{P}(\mathrm{B})=$
$0.7$ and $\mathrm{P}(\mathrm{A} \cap \mathrm{B})=0.2$, then $\mathrm{P}\left(\frac{\mathrm{B}}{\mathrm{A} \cup \overline{\mathrm{B}}}\right)$ is
(A) $\frac{3}{5}$
(B) $\frac{2}{5}$
(C) $\frac{1}{4}$
(D) $\frac{4}{5}$

Key Concepts

Conditional Probability

Conditional probability is the probability of an event occurring given that another event has already occurred. It is defined as P(B|E) = P(B ? E) / P(E) for any events B and E with P(E) > 0, and forms the basis of updating beliefs based on new information.

Complement of an Event

The complement of an event A, typically denoted ?, is the set of all outcomes in the sample space that are not part of A, with the property that P(A) + P(?) = 1. Understanding complements allows for determining the probability of an event by subtracting the probability of its complement from one.

Union and Intersection of Events

The union of two events represents all outcomes that are in either event, while the intersection represents the outcomes common to both events. These operations are fundamental in probability theory for combining events and using properties like P(A ? B) = P(A) + P(B) ? P(A ? B) to compute probabilities.

Transcript

00:01 Hello students our question is if two events a and b are such that probability of a bar is equal to 0 .4 probability of b is equal to 0 .7 and probability of a intersection b is equal to 0 .2 then probability of b given that a union b dash is equal to dash.

00:30 So here we have to find the probability of b given date a union b complement of b base so let's first time what write what is given to us we have given some quantities that is probability of a a days a day which is equal to which is equal to 0 .3 0 .3 probability of a dash is equal to 0 .3 probability of a dash is equal to 0 .3 so probability of a day is equal to 0 .3 so probability of a will be equal to probability of a is equal to one minus probability of a dash so which is equal to 0 .7 1 minus 0 .3 will be equal to 0 .7 and probability of b b is equal to 0 .7 so probability of b dash will be equal to 1 minus probability of b that is equal to 0.

01:36 Point and also the probability of a intersection b is given to us which is equal to 0 .2 so we have to find the probability of b given date a union b dash will be we have to find the probability of a union b -dish so let's start our solution to probability of b given date a union b -b -d -b -d -d - will be equal to probability of b intersection a union b -d -d -d -out of probability of probability of a union b this so this can be written as using distributive property offset we can write this as probability of probability of the intersection a union b intersection b intersection b union b intersection b dash if we use distributive property here so this will be equal to probability of b intersection a union probability of b intersection b dash.

03:29 So the probability of a union b dash can be written as probability of a plus probability of probability of b dash b dash minus probability of a intersection b dash.

03:57 Now this will be equal to as we know that a intersection b is equal to b intersection a.

04:09 These are the commutative property.

04:11 So b intersection b dash, the intersection of b with b dash will be equal to null set.

04:20 So here this will be equal to probability of a intersection b intersection b, intersection b union null set out of probability of a plus probability of b dash minus probability of a intersection b dash can be written as probability of a probability of a minus probability of a intersection intersection b this can be written as probability of a minus probability of a intersection beat so this is equal to the probability of a intersection b out of probability of a plus probability of b dash minus probability of a minus probability of a intersection b now this will be equal to putting the values.

05:55 We have probability of a intersection b...

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