PROBABILITY AND ODDS EXAMPLES

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 

  =  (⁵C₃ ⋅ ¹¹C₈) / ¹⁶C₁₁   --------(1)

Probability of getting 4 bowlers and 5 bats man 

  =  (⁵C₄ ⋅ ¹¹C₇) / ¹⁶C₁₁ --------(2)

Probability of getting 5 bowlers and 6 bats man 

  =  (⁵C₅ ⋅ ¹¹C₆) / ¹⁶C₁₁ --------(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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