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# Math Examples of Multiplication Theorems on Probability

• ## Four persons are chosen from a group of 3 men, 2 women and 4 children. Find the probability of getting four persons, where exactly two persons are children?

We use following steps for Probability of getting four persons, where exactly two persons are children -
Step 1: First of all, we find out Sample Space of given situation means here we find out how many persons are there-
3 + 2 + 4 = 9,
So, there are 9 persons.
Step 2: Now we calculate how many cases we have to retrieve four persons from there. So, total 9C4 cases because from 9 persons, we have to calculate 4 persons.
Therefore, there are total 9C4 cases.
Step 3: Now we find out how many ways is there for getting exactly two children from 4 persons-
4C2 are the total number of ways for evaluating an exactly 2 children from 4 person means
4C2.
So, there are 4C2 ways for getting an exactly 2 children from 4 persons.
Step 4: Now we find out how many ways is there for getting other 2 person from there-
5C2 are the total number of ways for evaluating other 2 persons from there means
5C2.
So, there are 5C2 ways for getting other two persons from there.
Step 5: Now we calculate probability of total number of favorable cases by using multiplication theorem on probability-
Total number of favorable cases ,
= number of ways to getting exactly 2 children * number of ways to getting other 2 person
4C2 * 5C2
So, total number of favorable cases is 4C2 * 5C2.
Now we calculate probability of getting four person, where exactly two children are there
= total favorable case / total sample space
= (4C2 * 5C2) / 9C4,
= 6 * 10 / 126,
= 10 / 21,
Therefore, probability of getting four person, where exactly two children is 10 / 21.

## Six cards are drawn at random from a pack of 52 cards. What is the probability that 3 cards are black and 3 cards are red?

We use following steps for getting the Probability of 3 red cards and 3 black cards-
Step 1: First of all, we find out Sample Space of given situation means here we find out how many cards are stored in a pack-
There are 52 cards are stored in a pack.
Step 2: Now we calculate how many cases we have to retrieve from given pack, there are 6 cards we have to retrieve in one time. So, there are total 52C6cases because in a pack, there are 52 cards and we have to retrieve 6 random cards from given pack.
Therefore, there are 52C6 total cases.
Step 3: Now we find out how many ways are there for getting 3 red cards from that pack-
26C3 are the total number of ways for evaluating 3 red cards from that pack because there are 26 red cards.
So, there are 26C3 ways for getting 3 red cards from that pack.
Step 4: Now we find out how many ways are there for getting 3 blue cards from that pack-
26C3 are the total number of ways for evaluating 3 blue cards from that pack because there are 26 blue cards.
So, there are 26C3 ways for getting 3 blue cards from that pack.
Step 5: Now we calculate probability of total number of favorable cases by using multiplication theorem on probability-
Total number of favorable cases,
= number of ways to getting 3 red card * number of ways to getting 3 blue card.
26C3 * 26C3,
So, total number of favorable cases is 26C3 * 26C3.
Now we calculate probability that two balls drawn are red and white
= total favorable case / total sample space,
26C3 * 26C3 / 52C6,
= 13000 / 39151,
Therefore, probability of getting 6 random cards, which have 3 red cards and 3 black cards are
13000 / 39151.

## A bag contains 3 red, 6 white and 7 blue balls. What is the probability that two balls drawn are red and white?

We use following steps for Probability of red and white balls -
Step 1: First of all, we find out Sample Space of given situation means here we find out how many balls are stored in a bag -
3 + 6 + 7 = 16,
So, there are 16 balls are stored in a bag.
Step 2: Now we how many cases we have to retrieve from given bag, there are 2 balls we have to retrieve in one time. So, total case are-
16C2 = (16 * 15) / (2 * 1),
= 120.
Therefore, there are 120 total cases.
Step 3: Now we find out how many ways is there for getting red ball in that bag-
3C1 are the total number of ways for evaluating a red ball in that bag means,
3C1 = 3,
So, there are 3 ways for getting a red ball from that bag.
Step 4: Now we find out how many ways is there for getting blue ball in that bag-
7C1 are the total number of ways for evaluating a red ball in that bag means.
7C1 = 7,
So, there are 7 ways for getting a red ball from that bag.
Step 5 : Now we calculate probability of total number of favorable cases by using multiplication theorem on probability-
Total number of favorable cases
= number of ways to getting red ball * number of ways to getting blue ball
= 3 * 7
= 21
So, total number of favorable cases is 21.
Now we calculate probability that two balls drawn are red and white
= total favorable case / total sample space
= 21 / 120
= 7 / 40
Therefore, probability of two balls drawn are red and white is 7 / 40.
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