Probability?

Question

a) In a sales campaign, for every item bought, shoppers are given a card with a picture of a film star on it. there are 10 different pictures and any shopper who collects a complete set of all 10 pictures wins a gift. on any occasion, a card recieved is equally likely to carry any one of the 10 pictures in the set. i) Find the prob that the 1st 4 cards received result in a shopper having exactly 3 different pictures.
ii) At a certain stage, a shopper has collected 9 out of the 10 pictures. find the least value of n such that P(at most n more cards are needed to complete the set) > 0.99
b) Susie has forgotten her pin number that consists of 4 digits in order. (The digits range from 0 - 9). She tries to recall her pin. Given that P(1st digit is correct) = 08., P(2nd digit is correct)= 0.86 and P(1st 2 digits are correct)=0.72. Find P(1st digit is incorrect and 2nd digit is incorrect).

Answer 1

a - i) A simple count shows:

1 10 (obvious)
2 630 (C(10,2) * (2^4 - 2) = 45 * 14)
3 4320 (P(10,3) * 6)
4 5040 (P(10,4) = 10*9*8*7)
-----
10000

so prob 3 distinct out of first 4 is 0.4320.

[For 2 distinct picture a < b (C(10,2) = 10! / (2! 8!) = 10*9/2 = 45) times count { aaaa, aaab, aaba, ..., bbab, bbbb } - count ={ aaaa, bbbb } = 2^4 - 2 = 16 - 2 = 14.]

[For 3 distinct picture a, b, c distinct in any order (P(10,3) = 10! / 3! = 10*9*8 = 720) times fitting in the duplicate count { aabc abac abca, abbc abcb, abcc } = 6.]

a-ii) prob still incomplete after k more cards is 0.9^k, which is less than 0.01 when k log(0.9) < log(0.01), i.e., k > approx. 43.7. So n = 44.

b) Draw the Venn diagram with area A = 0.8 [hopefully the intended correction of your typo of "08."], area = 0.86, area A and B = area overlap = 0.72. Then area A or B is 0.8 + 0.86 - 0.72 = 0.94. So its complement has area (probability) 1 - 0.94 = 0.06.