In a box there are 16 ice-creams: 6 lemon flavor,4 mint flavor and 6 strawberry flavor.When we extract two ice-creams,what's the probability of getting two different flavors,given that at least one is strawberry flavor.

That's my solution :

$P(A)$ = "Different flavors" = $\frac{{\textstyle (}\genfrac{}{}{0ex}{}{16}{2}{\textstyle )}}{16!}$

$P({B}^{c})$ = "At least one is strawberry flavor" = $1-\frac{{\textstyle (}\genfrac{}{}{0ex}{}{10}{2}{\textstyle )}}{16!}$

We want $P(A|{B}^{c})$ using conditional probability, where I go wrong?

That's my solution :

$P(A)$ = "Different flavors" = $\frac{{\textstyle (}\genfrac{}{}{0ex}{}{16}{2}{\textstyle )}}{16!}$

$P({B}^{c})$ = "At least one is strawberry flavor" = $1-\frac{{\textstyle (}\genfrac{}{}{0ex}{}{10}{2}{\textstyle )}}{16!}$

We want $P(A|{B}^{c})$ using conditional probability, where I go wrong?