Question

Question:

(a) Define the probability of an event when all outcomes are equally likely.

(b) What is the probability that you select the six winning numbers in a lottery if the six different winning numbers are selected from the first 50 positive integers?

Short answer

Answer

(a) The resultant answer is\(P(X = i) = \frac{1}{n}\).

(b) The resultant answer is\(\frac{1}{{15,890,700}} \approx 6.2930 \times {10^{ - 8}}\).

Step-by-step solution

Given data

The given datais first 50 positive integers.

Concept of Permutation 

Definition permutation (order is important):\(P(n,r) = \frac{{n!}}{{(n - r)!}}\)

Definition combination (order is not important):\(C(n,r) = \left( {\begin{aligned}{*{20}{l}}n\\r\end{aligned}} \right) = \frac{{n!}}{{r!(n - r)!}}\)

with \(n! = n \cdot (n - 1) \cdot \ldots \cdot 2 \cdot 1\).

Simplify the expression

(a)

Let there be \(n\) equally likely outcomes.

Then we have 1 chance in \(n\) to select one of the outcomes \(i\):\(P(X = i) = \frac{1}{n}\).

Find the probability of first 50 positive integer

(b)

The order of the 6 selected integers does not matter (because a different order will lead to the same numbers being drawn), thus we need to use the definition of a combination.

We need to select 6 numbers from 50.

\(\begin{aligned}{}n = 50\\r = 6\end{aligned}\)

Repetition is not allowed (since the selected numbers have to be different):

\(\begin{aligned}{}C(50,6) &= \frac{{50!}}{{6!44!}}\\C(50,6) &= 15,890,700\end{aligned}\)

1 of the 15,890,700 combinations will then lead to the winning combination:

The probability is the number of favorable outcomes divided by the number of possible outcomes:

\(\begin{aligned}{}P({\rm{ win }}) &= \frac{{\# {\rm{ of favorable outcomes }}}}{{\# {\rm{ of possible outcomes }}}}\\P({\rm{ win }}) &= \frac{1}{{15,890,700}}\\ \approx 6.2930 \times {10^{ - 8}}\end{aligned}\)