Here we are going to see some example problems on probability.

The word odds is frequently used in probability and statistics. Odds relate the chances in favour of an event A to the chances against it. Suppose a represents the number of ways that an event can occur and b represents the number of ways that the event can fail to occur.

The odds of an event A are a : b in favour of an event and

P(A) = a / (a + b)

Further, it may be noted that the odds are a : b in favour of an event is the same as to say that the odds are b : a against the event.

If the probability of an event is p , then the odds in favour of its occurrence are p to (1− p) and the odds against its occurrence are (1− p) to p .

**Example 1 :**

A cricket club has 16 members, of whom only 5 can bowl. What is the probability that in a team of 11 members at least 3 bowlers are selected?

**Solution :**

Number of members in a club = 16

= 11 batsman + 5 bowlers

At least 3 bowlers mean, we may select 3 bowlers, 4 bowlers and 5 bowlers.

- If we select 3 bowlers out of 5, we have to choose 8 bats man out of 11.
- If we select 4 bowlers out of 5, we have to choose 7 bats man out of 11.
- If we select 5 bowlers out of 5, we have to choose 6 bats man out of 11.

Probability of getting 3 bowlers and 8 bats man

= (^{5}C_{3 }⋅ ^{11}C_{8}) / ^{16}C_{11 } --------(1)

Probability of getting 4 bowlers and 5 bats man

= (^{5}C_{4 }⋅ ^{11}C_{7}) / ^{16}C_{11} --------(2)

Probability of getting 5 bowlers and 6 bats man

= (^{5}C_{5 }⋅ ^{11}C_{6}) / ^{16}C_{11} --------(3)

(1) + (2) + (3)

= (1650 + 1650 + 462) / 4368

= 3762/4368

= 627/728

**Example 2 :**

(i) The odds that the event A occurs is 5 to 7, find P(A).

(ii) Suppose P(B) = 2/5. Express the odds that the event B occurs.

**Solution :**

(i) P(A) = a/(a+b)

= 5/(5+7)

= 5/12

(ii) Suppose P(B) = 2/5. Express the odds that the event B occurs.

Note :

If the probability of an event is p , then the odds in favour of its occurrence are p to (1− p).

By using the point given in note, we may find the answer.

= (2/5) to 1 - (2/5)

= (2/5) to (3/5)

= 2 to 3.

So, the answer is 2 to 3.

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