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Sunday Afternoon Maths LXVICryptic crossnumber #2
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Square and cube endings
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Bending a straw
Two points along a drinking straw are picked at random. The straw is then bent at these points. What is the probability that the two ends meet up to make a triangle?
The sixth cent
You toss 6 fair coins, and I toss 5 fair coins. What is the probability that you get more heads than I do?
Source: UKMT Senior Maths Challenge 2014
A bag contains \(m\) blue and \(n\) yellow marbles. One marble is selected at random from the bag and its colour is noted. It is then returned to the bag along with \(k\) other marbles of the same colour. A second marble is now selected at random from the bag. What is the probability that the second marble is blue?
Timothy and Urban are playing a game with two six-sided dice. The dice are unusual: Rather than bearing a number, each face is painted either red or blue.
The two take turns throwing the dice. Timothy wins if the two top faces are the same color, and Urban wins if they're different. Their chances of winning are equal.
The first die has 5 red faces and 1 blue face. What are the colours on the second die?
The blue-eyed sisters
If you happen to meet two of the Jones sister (two sisters chosen at random from all the Jones sisters), it is exactly an even-money bet that both will be blue-eyed. What is your best guess of the total number of Jones sisters?
Can two (six-sided) dice be weighted so that the probability of each of the numbers 2, 3, ..., 12 is the same?
A knot of spectators in Downing Street was watching members of the Cabinet as they arrived for a critical meeting.
"Who's that?" I asked my neighbour, as a silk-hatted figure, carrying rolled umbrella, rang the bell at No. 10. "Is it the Minister of Maths?"
"Yes," he said.
"Quite right," said a second spectator. "The Minister of Maths it is. Looks grim, doesn't he?"
The first of the speakers tells the truth three times out of four. The second tells the truth four times out of five.
What is the probability that the gentleman in question was in fact the Minister of Maths?