Question

In one lottery, a player wins the jackpot by matching all five numbers drawn from white balls ( 1 through 45) and matching the number on the gold ballc( 1 through 34). If one ticket is purchased, what is the probability of winning the jackpot?

Answer 1

Answer:

1 in  41,539,806.

Step-by-step explanation:

The number of ways to pick 5 numbers from 45 = 45C5

= (45*44*43*42*41) / (5*4*3*2*1)

= 1,221,759.

There are 34 ways to pick a gold ball so total number of ways to get  6 matching numbers is

1221759*34

= 41,539,806.

Answer 2

Answer: $\dfrac{1}{41539806}$

Step-by-step explanation:

Given : Total white balls = 45

Number of correct balls needed to win = 5

Using combination ,

The total combination of getting 5 balls selected from 45 balls = $^{45}C_5=$

Total gold balls = 34

Similar;y , Number of ways to select 1 ball out of 34 = $^{34}C_1$

Total number of ways of selecting 5 white balls and 1 gold balls = $^{45}C_5\times^{34}C_1$

Using combination formula $^nC_m=\dfrac{n!}{m!(n-m)!}$

$=\dfrac{45\times44\times43\times42\times41\times40!}{120\times40!}\times\dfrac{34!}{1!(34-1)!}\\\\=1221759\times34=41539806$

i.e. Total number of possible ways of selecting 5 white balls and 1 gold balls =41539806

But there is only one combination for wining the jackpot.

So the probability of winning the jackpot= $\dfrac{\text{Favorable}}{\text{Total}}=\dfrac{1}{41539806}$

Hence, the required probability = $\dfrac{1}{41539806}$