A box contains 16 red, 12 white, and 15 yellow identical marbles. A man picks one marble at a time without replacement. How many times must he pick a marble to be 100% certain of picking at least 3 white marbles?
To solve this problem, we need to determine the minimum number of marbles that must be picked to ensure 100% certainty of having at least 3 white marbles. The total contents of the box are:
Total number of marbles = 16 + 12 + 15 = 43 marbles.
To ensure we have at least 3 white marbles, we must consider the worst-case scenario in which the maximum number of marbles picked are either red or yellow. This would mean picking all non-white marbles first.
Non-white marbles = 16 (red) + 15 (yellow) = 31 marbles.
If we pick all 31 non-white marbles, the next marble we pick (32nd) could be white. However, the task is to ensure we have at least 3 white marbles. Therefore, after picking all 31 non-white marbles, we need to pick an additional 3 marbles to ensure all are white, which gives us:
Total picks = 31 (non-white) + 3 (white) = 34 marbles.
This ensures with 100% certainty that we have at least 3 white marbles.
Conclusion: The minimum number of marbles the man must pick to be 100% certain of picking at least 3 white marbles is 34.