If a group included 70% men and 30% women, and 40% were left handed, what would be the probability of a randomly selected person being left handed?
I wonder if there is some piece of information (or a part of the question) that is missing here... As this question is asked, the probability of selecting a left handed person at random from this group will be 40 percent. Since 40 percent of all of the people are left handed, the fact that some are men and some are women seems to me to be irrelevant to this question.
If you are asking about the chances of selecting a left handed man or a left handed woman, that would be different, but if you are just asking for a left handed person, the answer is 40%.
When the percentage of left handed persons (including both men and women) in the group is 40% it means proportion of left handed persons in the group of 0.4 of the total persons. Therefor the probability of a randomly selected person from the group being left handed is 0.4. Please note that the percentage of men and and women in the group has no impact on the probability of the person being left handed.
However, if we want to calculate the probability of a randomly selected person being a left handed man, it can be done as follows.
The probability of the person selected being a male is 0.7. Further having selected a man the probability of the selected man being left handed is 0.4. Therefor probability of the selected person being left handed man
= 0.7 x 0.4 = 0.28
Similarly, probability of the selected person being left handed woman
= 0.3 x 0.4 = 0.12