Probability of Three of a Kind

Probability is the chance of the outcome of an event of a particular experiment. An event is a one or more possible outcomes of an experiment. Probabilities are occurs always between impossible
and possible. The set of all possible outcomes of a particular experiment is called as sample space.

P(Event) = Number of possible ways an event occurs / Total number of possible outcomes.

For example, tossing a fair coin, the possible outcomes are {head, tail} = 2. So, probability of occurs an event head is `1/2` .

Probability of three of a Kind - Example Problems

Ex 1: A box has 10 apples and 10 bananas. If three fruits are drawn at random, find the probability that they will be of same kind.

Sol:

Let S be the sample space, n(S) = 10 +10 = 20.

We have to select 3 fruits from 20 fruits.

n(S) = C(20, 3) = `(20!)/(3!xx17!) ` = `(20xx19xx18)/(3xx2xx1)` = 1140

A be the event of getting 3 apples

B be the event of getting 3 bananas

n(A) = C(10, 3) = `(10!)/(3!xx7!) ` = `(10xx9xx8)/(3xx2xx1)` = 120

n(B) = C(10, 3) =` (10!)/(3!xx7!) ` = `(10xx9xx8)/(3xx2xx1)` = 120

P(A) =` (n(A))/(n(S))` = `120/1140`

P(B) = `(n(B))/(n(S))` = `120/1140`

P(A or B) = P(A) + (B) = `120/1140` + `120/1140` =`4/19`

P(3 apples or 3 bananas) = `4/19` .

Ex 2: A box has 12 red and 14 black marbles. If three marbles are drawn at random, what is the probability that they will be of same kind?

Sol:

Let S be the sample space, n(S) = 12 +14 = 26.

We have to select 3 marbles from 26 marbles.

n(S) = C(26, 3) = `(26!)/(3!xx23!) ` = `(26xx25xx24)/(3xx2xx1)` = 2600

A be the event of getting 3 red marbles

B be the event of getting 3 black marbles

n(A) = C(12, 3) = `(12!)/(3!xx9!) ` = `(12xx11xx10)/(3xx2xx1)` = 220

n(B) = C(14, 3) =` (14!)/(3!xx11!) ` = `(14xx13xx12)/(3xx2xx1)` = 364

P(A) =` (n(A))/(n(S))` = `220/2600`

P(B) = `(n(B))/(n(S))` = `364/2600`

P(A or B) = P(A) + (B) = `220/2600` + `364/2600` =`73/325`

P(3 red or 3 black marbles) = `73/325` .

Probability of three of a Kind - Practice Problems

Pr 1: A box has 8 apples and 12 bananas. If three fruits are drawn at random, find the probability that they will be of same kind.

Pr 2: A box has 6 red and 12 black marbles. If three marbles are drawn at random, what is the probability that they will be of same kind?

Ans: 1) `23/95` 2) `5/17`