Answer

Verified

414.9k+ views

**Hint:**Probability is a measure of the likelihood that an event will happen.

\[Probability{\text{ }} = \;\dfrac{{favorable\;outcomes}}{{possible\;outcomes}}\]

A conditional probability is the probability of one event if another event occurred. In the “die-toss” example, the probability of event A, three dots showing, is P (A) = \[\dfrac{1}{6}\] on a single toss. But what if we know that event B, at least three dots showing, occurred? Then there are only four possible outcomes, one of which is A. The probability of A = {3} is\[\dfrac{1}{4}\], given that B = {3, 4, 5, 6} occurred. The conditional probability of A given B is written P $\left( {\dfrac{A}{B}} \right)$.

The formula of conditional probability is:

$P\left( {\dfrac{A}{B}} \right) = \dfrac{{P\left( {A\bigcap B } \right)}}{{P(B)}}$

Event A is independent of B if the conditional probability of A given B is the same as the unconditional probability of A.

**Complete step-by-step answer:**

A black and a red dice are rolled.

Total possible outcomes =${6^2} = 36$………………… (1)

Let the A be an event obtaining a sum 8

$

\Rightarrow A = \left\{ {\left( {2,6} \right),\left( {3,5} \right),\left( {5,3} \right),\left( {4,4} \right),\left( {6,2} \right)} \right\} \\

\Rightarrow n\left( A \right) = 5 \\

$……………………… (2)

B be an event that the red die resulted in a number less than 4.

\[

\Rightarrow B =

\left\{

\left( {1,1} \right),\left( {1,2} \right),\left( {1,3} \right), \\

\left( {2,1} \right),\left( {2,2} \right),\left( {2,3} \right), \\

\left( {3,1} \right),\left( {3,2} \right),\left( {3,3} \right), \\

\left( {4,1} \right),\left( {4,2} \right),\left( {4,3} \right), \\

\left( {5,1} \right),\left( {5,2} \right),\left( {5,3} \right), \\

\left( {6,1} \right),\left( {6,2} \right),\left( {6,3} \right) \\

\right\} \\

\Rightarrow n\left( B \right) = 18 \\

\] ……………………………….(3)

Let’s find intersect when both event A and B happens by equation (2) and (3):-

$

\Rightarrow A\bigcap B = \left\{ {\left( {5,3} \right),(6,2)} \right\} \\

\Rightarrow n\left( {A\bigcap B } \right) = 2 \\

$

Probability of $A\bigcap B $= \[\dfrac{{n\left( {A\bigcap B } \right)}}{{total\;possible\;outcomes}} = \dfrac{2}{{36}}..................(1)\]

Probability of event B = P (B) =$\dfrac{{18}}{{36}}$………………………….(2)

Using the formula mentioned in the hint of conditional probability and putting the values of equation (1) and (2).

$

P\left( {\dfrac{A}{B}} \right) = \dfrac{{P\left( {A\bigcap B } \right)}}{{P(B)}} \\

\Rightarrow P\left( {\dfrac{A}{B}} \right) = \dfrac{{\dfrac{2}{{36}}}}{{\dfrac{{18}}{{36}}}} \\

\Rightarrow P\left( {\dfrac{A}{B}} \right) = \dfrac{1}{9} \\

$

The conditional probability of obtaining the sum 8, given that the red die resulted in a number less than 4 is $\dfrac{1}{9}$

**Note:**Conditional probability is used in many areas, in fields as diverse as calculus, insurance, and politics. For example, the re-election of a president depends upon the voting preference of voters and perhaps the success of television advertising—even the probability of the opponent making gaffes during debates!

Recently Updated Pages

Differentiate between Shortterm and Longterm adapt class 1 biology CBSE

How do you find slope point slope slope intercept standard class 12 maths CBSE

How do you find B1 We know that B2B+2I3 class 12 maths CBSE

How do you integrate int dfracxsqrt x2 + 9 dx class 12 maths CBSE

How do you integrate int left dfracx2 1x + 1 right class 12 maths CBSE

How do you find the critical points of yx2sin x on class 12 maths CBSE

Trending doubts

Which are the Top 10 Largest Countries of the World?

Define limiting molar conductivity Why does the conductivity class 12 chemistry CBSE

Give 10 examples for herbs , shrubs , climbers , creepers

Difference Between Plant Cell and Animal Cell

Fill the blanks with the suitable prepositions 1 The class 9 english CBSE

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

Name 10 Living and Non living things class 9 biology CBSE

The Buddhist universities of Nalanda and Vikramshila class 7 social science CBSE

Write a letter to the principal requesting him to grant class 10 english CBSE