IB Math HL – Binomial Distribution

IB Maths HL 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 ( $X\sm B(n,p)$ ) 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 HL – Discrete Probability Distribution, Binomial Distribution

How can we find the probability P(X=3) when $X\sim B(6,0.3)$

Solution

IB Mathematics HL – Discrete Probability Distribution, Binomial Distribution

$P(X=3)= \displaystyle \binom{6}{3} (0.3)^3 (0.7)^{6-3}=0.18522$

Using GDC Casio fx-9860G SD

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

Setting Data: Variable

x: 3

Numtrial:3

p: 0.3

Execute

We find that the probability is

$P(X=3)= 0.18522$

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

How can we find the probability P(X\geq 3) when $X\sim B(7,0.2)$

Solution

IB Mathematics HL – Discrete Probability Distribution, Binomial Distribution

$P(X\geq 3) = P(X=3)+P(X=4)+P(X=5)+P(X=6)+P(X=7)=$

$=1-P(X\leq 2)=1-P(X=0)-P(X=1)-P(X=2)= 0.148032$

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 $P(X\leq 2) = 0.851968$

Therefore, $P(X\geq 3) =1-P(X\leq 2) = 1-0.851968=0.148032$