There are five Power Rangers. Of the five, two are male(Red, Black) and three are female (Pink, Yellow,blue.)I randomly select two of the Power Rangers, without replacement.
(a) What is the probability that the black Ranger is one of the two selected Power Rangers?
pretending that you only randomly select one power ranger, the
probability of picking the black would just be 1/5, so if you were
to pick another it would be 1/4 so it would be 1/5 * 1/4 = .05 so 5% chance?
it just seems low
(b) What is the probability at least one of the two selected Power Rangers is female?
probability that one is female is 3/5, so if you were to pick
another power ranger at random would it be 3/5*3/4
(c) Given that at least one of the two selected Power Rangers is male, what is the conditional probability the Pink Ranger is selected?
conditional probability, wouldnt it just be 33.3% that the pink is selected given theres only three females?
(d) Are the events “the Pink Ranger is one of the two selected” and “at least one of the two selected Power Rangers is female” independent? Explain why or why not.
Yes, event A effects event B.
Best Answer
As the commenters have pointed out, your intuition to doubt your answer to (a) was appropriate. $$ P(\text{selected Pink}) = P(\text{Pink selected first}) + P(\text{Not pink, then Pink}) = 1/5 + 4/5 \times 1/4 = 2/5. $$
Then $$ P(\text{At least } 1 \text{ female}) = 1 - P(\text{both male}) = 1 - 3/5 \times 2/4 = 1 - 3/10 = 7/10. $$ Next, use Bayes's Theorem: $$ P(\text{Pink}|\text{At least 1 female}) = \frac{P(\text{Pink AND at least 1 female})}{P(\text{At least 1 female})} = \frac{P(\text{Pink})}{P(\text{At least 1 female})} = \frac{2/5}{7/10} = \frac{4}{7}. $$ (Can you see why the middle equality is true? This is useful in the final part!)
Finally, recall that two events $A,B$ are independent if $P(A \cap B) = P(A)P(B)$. Can you use this to give a clearer explanation for (d)?