15. For each of the following, write the probability as the intersection of two events. Then, indicate whether the two events are independent or dependent, and calculate the probability of
the intersection of the two events occurring.
a. The probability of selecting a 6 from the first draw and a 7 on the second draw when two balls are selected without replacement from a container with 10 balls numbered 1 to 10
b. The probability of selecting a 6 on the first draw and a 7 on the second draw when two balls are selected with replacement from a container with 10 balls numbered 1 to 10
c. The probability that two people selected at random in a shopping mall on a very busy Saturday both have a birthday in the month of June. Assume that all 365 birthdays are equally
likely, and ignore the possibility of a February 29 leap-year birthday.
d. The probability that two socks selected at random from a drawer containing 10 black socks and 6 white socks will both be black

a. The probability of selecting a 6 from the first draw and a 7 on the second draw when two balls are selected without replacement from a container with 10 balls numbered 1 to 10

Not independent because without replacement

Prob for both = [tex]\frac{1}{10} \frac{1}{9} \\=\frac{1}{90}[/tex]

b. The probability of selecting a 6 on the first draw and a 7 on the second draw when two balls are selected with replacement from a container with 10 balls numbered 1 to 10

Here independent because with replacement makes probability independent.

Prob for both = P(A) *P(B) = [tex]\frac{1}{10} *\frac{1}{10} \\=\frac{1}{100}[/tex]d

c. The probability that two people selected at random in a shopping mall on a very busy Saturday both have a birthday in the month of June. Assume that all 365 birthdays are equally

likely, and ignore the possibility of a February 29 leap-year birthday.

Here independent because one person birthday will not affect the other person birthday

Prob for both = [tex](\frac{30}{365})^2\\ =\frac{25}{5329}[/tex]

d. The probability that two socks selected at random from a drawer containing 10 black socks and 6 white socks will both be black

Prob for I sock black = 10/16 and II sock black if first sock is black = 9/15

Hence not independent

Prob for both = [tex]\frac{10C2}{16C2} \\=\frac{3}{8}[/tex]