Name: Addition Theorem of Probability: Illustrations
Uploaded: 2019-07-19
Duration: 3 min 27 s
Description: If A and B are two events associated with a random experiment, how do we find the probability of at least one of them? Let's watch and find out in this video.

Question

Set A : {randomly choosen

100

years in which

76

are simple and

24

are leap years}

Set B : {randomly choosen $100$ years in which $75$ are simple and $25$ are leap years}
An year is chosen from either set $A$ or set $B$ and is found to have $53$ sundays. Probability that the chosen year was a leap year.

Accepted Answer

Given:

Set $A$ : ${$ randomly chosen $100$ years in which $75$ are simple & $25$ are leap gear $}$

Set $B$ : $53$ randomly chosen for years in which $75$ are simple and $25$ are leap years $}$

$E_{1} \to$ Selection of leap years from set $A$ .

$∴ P (E_{1}) = \frac{24}{100}$

$E_{2} \to$ Selection of leap years from set $B$ .

$∴ P (E_{2}) = \frac{25}{100}$

$E_{2} \to$ Selection of simple year from set A.

$∴ P ({E_{1}}^{'}) = \frac{76}{100}$

${E_{2}}^{'} \to$ Selection of simple year from set B

$∴ P ({E_{2}}^{'}) = \frac{75}{100}$

$E \to$ Selected year has $53$ Sundays.

$∴$ When selected year is leap year then Probability of $53$ Sundays. $P (\frac{E}{E_{1}}) = \frac{2}{7} = P (\frac{E}{E_{2}}) .$

When selected year is simple year then probability of $53$ Sundays $= \frac{1}{7} = P (\frac{E}{E_{1}'}) = P (\frac{E}{E_{2}'})$

$∴ P (\frac{S e l e c t i o n o f l e a p y e a r}{S e l e c t e d y e a r h a s 53 s u n d a y s})$

$= \frac{P (E_{1}) . P (\frac{E}{E_{1}}) + P (E_{2}) . P (\frac{E}{E_{2}})}{P (E_{1}) . P (\frac{E}{E_{1}}) + P (E_{2}) . P (\frac{E}{E_{2}}) + P ({E_{1}}^{'}) . P (\frac{E}{E_{1}'}) + P ({E_{2}}^{'}) . P (\frac{E}{E_{2}'})}$

$= \frac{\frac{24}{100} \times \frac{2}{7} + \frac{25}{100} . \frac{2}{7}}{\frac{24}{100} . \frac{2}{7} + \frac{25}{100} . \frac{2}{7} + \frac{76}{100} . \frac{1}{7} + \frac{75}{100} . \frac{1}{7}}$

$= \frac{98}{249}$

Important Questions on Probability

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Set A : {randomly choosen 100 years in which 76 are simple and 24 are leap years} Set B : {randomly choosen 100 years in which 75 are simple and 25 are leap years} An year is chosen from either set A or set B and is found to have 53 sundays. Probability that the chosen year was a leap year.

Important Questions on Probability

Learn and Prepare for any exam you want !