Question

this is how the game is played. In a Powerball play slip, a player picks 5 numbers from 1 through 69 and 1 number from 1 through 26 (this is the Powerball number). The grand prize is awarded to the player (or players) whose ticket matches all of the numbers on the five chosen white balls and the one chosen red ball. What are the odds of winning? You need to calculate the odds of getting the exact 5 white balls and the odds of getting the Powerball number.

Answer 1

Solution

Back-up Theory

Number of ways of selecting r things out of n things is given by ⁿC_r = (n!)/{(r!)(n - r)!}……(1)

Values of ⁿC_r can be directly obtained using Excel Function: Math & Trig COMBIN……. (1a)

Now to work out the solution,

Vide (1), 5 numbers from 1 through 69 can be picked up in ⁶⁹C₅ ways

= 11238513 ways [Vide (1a)]

Of these, only one would be the winning combination. Hence,

P(ticket matches all of the numbers on the five chosen white balls) = (1/11238513)

Similarly, 1 number from 1 through 26 (this is the Powerball number) can be picked up in ²⁶C₁

= 26 ways [Vide (1)]

Again, only one would be the winning number. Hence,

P(ticket matches one chosen red ball) = (1/26).

Thus,

The odds of winning

= (the odds of getting the exact 5 white balls) x (the odds of getting the Powerball number)

= (1/11238513) x (1/26)

= 1 : 292201338 Answer

[or in probability language, 0.0000000034]

DONE