IB Math SL – Binomial Distribution

IB Maths SL Binomial Distribution

The Binomial distribution can be used in situations in which a given experiment (trial) is repeated a number of times. For the binomial model to be applied the following four criteria must be satisfied

– the trial is carried out a fixed number of times n.

– the outcomes of each trial can be classified into two ‘types’ conveniently named success or failure.

– the probability p of success remains constant for each trial.

– the individual trials are independent of each other.

If a discrete random variable X follows a binomial distribution ( $X\sim B(n,p)$ ) with n is the number of trials and p the probability of a success, then the probability distribution function is given by the following formula:

$P(X=x)= \displaystyle \binom{n}{x} p^x (1-p)^{n-x}, x=0,1,2,…,n$

Questions and Answers

1. IB Mathematics SL – Discrete Probability Distribution, Binomial Distribution

How can we find the probability P(X=2) when $X\sim B(5,0.3)$

Solution

$P(X=2)= \displaystyle \binom{5}{2} (0.3)^2 (0.7)^{5-2}=0.3087$

Using GDC Casio fx-9860G SD

MAIN MENU > STAT>DIST(F5)>BINM>Bpd>

Setting Data: Variable

x: 2

Numtrial:5

p: 0.3

Execute

We find that the probability is

$P(X=3)= 0.3087$

2. IB Mathematics SL – Discrete Probability Distribution, Binomial Distribution

How can we find the probability P(X\geq 2) when $X\sim B(8,0.2)$

Solution

$P(X\geq 2) =1-P(X\leq 1)=1-P(X=0)-P(X=1) = 0.49668353$

Using GDC Casio fx-9860G SD

MAIN MENU > STAT>DIST(F5)>BINM>Bcd>

Setting Data: Variable

x: 2

Numtrial:7

p: 0.2

Execute

We find the probability $P(X\leq 2) = 0.50331647$

Therefore $P(X\geq 3) =1-P(X\leq 2) = 1-0.50331647=0.49668353$