Question

There are 27 tickets in the lottery, of which 5 tickets win and 3 give the right to draw the next ticket. Calculate the probability of winning by purchasing one lottery ticket.

Answer 1

Number of tickets = 27

Number of tickets that win you the lottery = 5

Number of tickets that give you the right to draw the next ticket = 3

There are 4 cases to this problem -

• Case I : If one draws the winning ticket in the purchase

Probability of drawing the winning ticket = 5C1 / 27C1 = 5/27 = 0.18519

• Case II : If one draws the ticket that gives you the right to draw the next ticket and in the second draw one draws the winning ticket

Probability of drawing a ticket that gives you the right to draw the next ticket = 3C1 / 27C1 = 3/27 = 1/9

Probability of drawing the winning ticket = 5C1 / 26C1 = 5/26

Total probability of winning the lottery = (1/9) x (5/26) = 0.02136

• Case III : If one gets the ticket , that gives you the right to draw the next ticket and in the second draw also one draws the same type of ticket and in the third draw one draws the winning ticket

Probability of drawing a ticket that gives you the right to draw the next ticket = 3C1 / 27C1 = 3/27 = 1/9

Probability of drawing the same type of ticket in the second draw = 2C1 / 25C1 = 2/25

Probability of drawing the winning ticket = 5C1 / 24C1 = 5/24

Total probability of winning the lottery = (1/9) x (2/25) x (5/24) = 0.00185

• Case IV : If one gets the ticket , that gives you the right to draw the next ticket and in the second draw also one gets the same ticket and in the next draw also one draws the same type of ticket and in the next draw one draws the winning ticket

Probability of drawing a ticket that gives you the right to draw the next ticket = 3C1 / 27C1 = 3/27 = 1/9

Probability of drawing the same type of ticket in the second draw = 2C1 / 25C1 = 2/25

Probability of drawing the same type of ticket in the third draw = 1 / 24C1 = 1/24

Probability of drawing the winning ticket = 5C1 / 23C1 = 5/23

Total probability of winning the lottery = (1/9) x (2/25) x (1/24) x (5/23) = 0.00008

Therefore ,

Total probability of winning the lottery by purchasing one ticket = 0.18519 + 0.02136 + 0.00185 + 0.00008

= 0.20848

(NOTE THAT : The above probabilities are calculated when the tickets are drawn WITHOUT REPLACEMENT procedure)