Unizor - Probability Examples - Lottery

Here are the problems and answers. Solutions are presented on UNIZOR.COM (in the lecture and its notes) Pick 6 out of 49 Six winning numbers are randomly picked from a set of numbers from 1 to 49. You have to guess these picked numbers by filling a ticket prior to drawing the winning numbers. What is the probability of guessing exactly K numbers out of six winning (K is assumed to be in the interval from 0 to 6 inclusive)? Answer: C(6,K)·C(43,6−K)/C(49,6) Numerically, K=0: 0.43596498 K=1: 0.41302660 K=2: 0.13237803 K=3: 0.01765040 K=4: 0.00096862 K=5: 0.00001845 K=6: 0.00000007 Two Lottery Tickets The same rules of the lottery - six winning numbers are randomly drawn from a set of numbers from 1 to 49. You have to predict these picked numbers. But now you buy two tickets and mark your selection of six numbers on each with a condition that all numbers on these two tickets are different (so, 12 different numbers are represented on two tickets). What is the probability of none of these two tickets correctly selecting two or more winning numbers? That is, since correctly guessing two or more numbers on a ticket results in some pay-off, we are calculating the probability of complete loss of the cost of two tickets. Answer: [C(37,6)+12·C(37,5)+36·C(37,4)]/C(49,6) ≅ 0.7103 Minimum Number of Tickets The same rules of the lottery - six winning numbers are randomly drawn from a set of numbers from 1 to 49. You have to predict these numbers by filling the tickets. There is no restriction for choosing numbers on different tickets, you can repeat them in different combinations. To win, a ticket must have at least two winning numbers. What is the minimum number of tickets you have to buy to reduce the probability of total loss of all money spent on tickets (that is, to reduce the probability of each and every ticket to have only zero or one winning number) to below 0.5=50% level? Answer: 5 tickets.