Probability Distributions

Question

In a box of smarties there are eight different colours which normally occur in equal proportions. Rachael is given 24 smarties, and orange ones are her favourite. Assume they come from a very large box.

(i) How many orange ones would she expect to get?

(ii) What is the probability that she gets this number?

(iii) What is the probability that she gets fewer orange ones?

(iv) What is the probability that she gets more orange ones than she expected?

Answer 1

p = 1/8 for each color

i) E(x) = np= 24*1/8 = 3

ii) If we want to use the binomial distribution formula, P(x) = C(24,3)(1/8)³(7/8)²¹ =

2024 * .001953 * .060558 = .239392

Alternatively, use statistical functions. For example, in Excel, =binom.dist(3,24,1/8,0) = .239392

iii) P(X <= 3) we can calculate in two ways. We can calculate P(0) + P(1) + P(2) using the formula shown in ii:

C(24,0)(1/8)⁰(7/8)²⁴ + C(24,1)(1/8)¹(7/8)²³ + C(24,2)(1/8)²(7/8)²² = in Excel,

=COMBIN(24,0)*(1/8)^0*(7/8)^24 + COMBIN(24,1)*(1/8)^1*(7/8)^23 + COMBIN(24,2)*(1/8)^2*(7/8)^22 =

.408173

However, it is much easier in Excel to use the binom.dist shown above:

=binom.dist(2,24,1/8,1) = .408173. The final 1 means cumulative rather than just the probability of being that value as in ii)

iv) P(X > 3) = 1 - P(X <= 3) = 1 - the answers in ii and iii = 1 - .239392 - .408173 = .352435.

Alternatively, we can do 1 - binom.dist(3,24,1/8,1) = .352435.