Paper 4: Fundamentals of Business Mathematics & Statistic

FUNDAMENTALS OF BUSINESS MATHEMATICS AND STATISTICS I 10.5

Example 3 :
Four coins are tossed simultaneously. What is the probability of getting (a) 2 heads and 2 tails (6) at least
two heads (c) at least one head.
Solution :

P(x)=

4 114

22

xx

Cx

⎛⎞⎛⎞−

⎜⎜⎜⎜⎜⎜⎟⎟⎟⎟⎟⎟⎟⎟

⎝⎠⎝⎠

(a) Putting x = 2

P(2) =

4 22

2

11

C 22

⎛⎞⎛⎞⎜⎜⎟⎟⎟⎟

⎜⎜⎜⎜⎝⎠⎝⎠⎟⎟⎟⎟

=

4

4 2 1

C 2

⎛⎞⎜ ⎟⎟

⎜⎜⎝⎠⎟⎟

=^38

(b) At least 2 heads mean minimum 2 or 3 or 4 heads. Therefore, the probability :

P(≥2) = P 2 + P 3 + P 4

=

444444

234

111

CCC 222

⎛⎞ ⎛⎞ ⎛⎞⎜⎜⎜⎟⎟⎟⎟⎟⎟++

⎜⎜⎜⎜⎜⎜⎝⎠ ⎝⎠ ⎝⎠⎟⎟⎟⎟⎟⎟

= 311 11 841616 ++ =

(c) P = 1 – P (no head)

= 1 –

4 4

0

111115

C 21616

−=−=⎛⎞⎜ ⎟⎟

⎜⎜⎝⎠⎟⎟

10.2.2. PROPERTIES OF BINOMIAL DISTRIBUTION

1. Binomial distribution has two parameters – n and p (or q)


2. Mean = np


3. Variance = npq


4. Standard Deviation = npq


5. Binomial distribution is symmetrical if p = q = 0.5

Example 4 :
Five coins are tossed 3,200 times, find the frequencies of the distribution of heads and tails and tabulate the
results.
Solution :
p = 0.5 and q = 0.5
Applying binomial distribution, the probability of getting X beads is given by: