How to Solve Conditional and Binomial Probabilities?

Related Topics

• How to Solve Probability Problems?

A step-by-step guide to conditional and binomial probabilities

The conditional probability of an event is the probability that the event will occur, given that event A has already occurred. This probability can be written as \(P(B|A)\). The formula for conditional probability is:

\(\color{blue}{P\left(B│A\right)=\:\frac{P\:\left(A\:and\:B\right)}{P\left (A\right)}}\)

Where \(P\left(A\right)\ne 0\), which can be written as follow:

\(\color{blue}{P\left(B│A\right)=\:\frac{P\left(A∩B\right)}{P\left(A\right)}}\)

The binomial probability is called the probability of exactly x successes on n repeated trials in an experiment that has two possible outcomes, (where \(0\le x\le n\), \(p\) is the probability of success, and \(1-p\) is the probability of failure.) and is obtained by the following formula:

\(\color{blue}{P=\:_nC_x\:\:p^x\:\left(1-p\right)^{n-x}}\)

Conditional and Binomial Probabilities – Example 1:

A bag contains \(3\) blue, \(5\) red, and \(10\) white marbles. Two marbles are chosen at random. Given that the first marble is red, find the probability that the second marble is blue. (Assume that the first marble has not been replaced.)

Solution: Use the conditional probability formula to solve this problem: \(P\left(B│A\right)=\:\frac{P\:\left(A\:and\:B\right)}{P\left (A\right)}\)

The probability of selecting red marble: \(P (red)=\frac{5}{3+5+10}=\frac{5}{18}\)

The probability of selecting blue marble: \(P (blue)=\frac{3}{3+4+10}=\frac{3}{17}\)

The probability of selecting red and blue marble: \(P (red and blue)=\frac {5}{18}×\frac{3}{17}=\frac{5}{102}\)

Therefore,\(\:P\left(blue│red\right)\)\(=\frac{P\:\left(red\:and\:blue\right)}{P\:\left(red\right)}=\frac{\frac{5}{102}}{\frac{5}{18}}=\frac{3}{17}\)