Question

Suppose that a computer randomly generates four-digit codes to match ATM pin codes among {0000, 0001, . . . , 9999}.

1. Explain carefully why there are 10000 possible four-digit codes when the digits are among 0-9.

2. Note that 10000 = 104. Explain how this connects the number of digits, the choice of digits, and the possible four-digit pin codes.

3. Among the 10000 possible four-digit pin codes, how many involve exactly three 1s or exactly three 2s? Write out the possibilities.

4. Using the previous part, what is the probability that a randomly selected four-digit pin code will contain exactly three 1s or exactly three 2s?

5. Using the complement rule, what is the probability that a randomly selected four-digit pin code will not contain exactly three 1s or exactly three 2s?

Answer 1

(a)

each digit can be filled by any one of the number {0,1,2,3,4,5,6,7,8,9}

so each digit has 10 possibility and our code has 4 digit

Number of possibe four-digit code=10*10*10*10=10000

(b)

number of codes having exactly three 1s or exactly three 2s are=36+36=72

0111,2111,3111,4111,5111,6111,7111,8111,9111,

1011,1211,1311,1411,1511,1611,1711,1811,1911,

1101,1121,1131,1141,1151,1161,1171,1181,1191,

1110,1112,1113,1114,1115,1116,1117,1118,1119

0222,1222,3222,4222,5222,6222,7222,8222,9222,

2022,2122,2322,2422,2522,2622,2722,2822,2922,

2202,2212,2232,2242,2252,2262,2272,2282,2292,

2220,2221,2223,2224,2225,2226,2227,2228,2229