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56 problems match “probability”. Clear

Every month has a birthday ●●●○○ 50 ptsAssume 40 people are independent and each birthday is equally likely to fall in any of the 12 months. What is the probability that every one of the 12 months contains at least one of the 40 birthdays? Give your answer as a decimal rounded to seven decimal places.
Color of the last ball ●●●○○ 50 ptsMilan has a bag of 2020 red balls and 2021 green balls. He repeatedly draws 2 balls out of the bag uniformly at random. If they are the same color, he changes them both to the opposite color and returns them to the bag. If they are different colors, he discards them. Eventually the bag has 1 ball left. Let $p$ be the probability that it is green. Compute $\lfloor 2021 p \rfloor$ .
Smallest group with matching last four digits ●●●○○ 50 ptsAssume each person's last four digits are equally likely to be any of 0000 to 9999, independently. What is the smallest number of people needed so that the probability that at least two of them have the same last four digits is at least 99.9%?
Red balls after discarding draws ●●●○○ 50 ptsAn urn contains 2 yellow balls, 3 blue balls and 22 red balls. One ball is drawn, found to be red, and discarded. Then 5 balls are drawn and thrown away without looking at their colours. Given this information, what is the probability that if you now draw four balls from the urn, they are all red? Round to three decimal places.
Ten numbered coins sum at least 45 ●●●○○ 50 ptsYou have 10 coins. Coin $i$ shows the number $i$ on one side (for $i = 1, \dots, 10$ ) and $0$ on the other side. You toss all the coins and add up the values that face up. Find the probability that the sum is greater than or equal to $45$ . Give your answer as a decimal.
Non-decreasing rolls of a die ●●●○○ 50 ptsA fair die is tossed 6 times. What is the probability that on each toss, the number shown is greater than or equal to the number on the previous toss? Give your answer as a decimal rounded to four decimal places.
Ten ten-sided dice sum above 95 ●●●○○ 50 ptsTen fair dice are rolled together, each with equally likely faces numbered 1 through 10. Find the probability that the total of the ten top faces is greater than 95. Give your answer as a decimal.
Matching head counts with unfair coin ●●●○○ 50 ptsJackie and Phil have two fair coins and a third coin that comes up heads with probability $\frac{4}{7}$ . Jackie flips the three coins, and then Phil flips the three coins. Let $\frac{m}{n}$ be the probability that Jackie gets the same number of heads as Phil, where $m$ and $n$ are relatively prime positive integers. Find $m + n$ .
Each player gets an ace ●●●○○ 50 ptsA standard 52-card deck is shuffled and 2 cards are dealt to each of 4 players. Find the probability that every player receives at least one ace. Give your answer as a decimal rounded to eight decimal places.
One ace per suit in pinochle ●●●○○ 50 ptsA player is dealt a 20-card hand from a double-deck pinochle deck of 80 cards, which contains 4 aces of each suit. Find the probability that the hand contains exactly 1 ace of each of the four suits. Round to four decimal places.