Exercise 1.3. a telephone number consists of ten digits, of which the first digit is one of 1; 2; : : : ; 9 and the others can be 0; 1; 2; : : : ; 9. what is the probability that 0/ appears at most once in a telephone number, if all the digits are chosen completely at random

The probability that "0" appears at most once in a telephone number is 0.82

Step-by-step explanation:

Telephone number = 10 digits

First digit = (1,2......9)------> there are 9 possible digits

digit 2 - digit 10 = (0,1,2......9) -----> there are 10 possible digits

Let's find the total amount of telefone numbers with those characteristics:

possible first digit X possible second digit X...............X possible tenth digit

9X10X10X10X10X10X10X10X10X10 =

Let's find the amount of telephone numbers that "0" appears at most once (it means 0 will occur either once or will not occur):

If "0" appears once, the rest of the numbers should be 1.2.....9 (9 possible digits)

9X10X9X9X9X9X9X9X9X9 =

If "0" doesn't appear, the ten digits should be 1,2....9

9X9X9X9X9X9X9X9X9X9 =

Finally, find the probability

P=(favorable outcomes/total outcomes)

P= = 0.82

we have three similar triangles, because each has a right angle and shares an angle.   let's write the angles in order: opposite to short leg, long leg, hypotenuse.

cab similar to bad similar to cbd

or as ratios,

ca: ab: cb = ba: ad: bd = cb: bd: cd

we also know

ac = ad + cd

(ad+cd): ab: cb = ba: ad: bd = cb: bd: cd

(ad+cd)/cb=cb/cd

we have cb=6, ad=5 and seek x=cd.

we reject the negative root and conclude x=4

answer: 4