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6 problems match “geometric distribution”. Clear

Prime sum of white and red balls ●●○○○ 20 ptsA bag contains 6 red balls and 8 white balls. Five balls are taken at random and placed in box $A$ , and the remaining 9 balls are placed in box $B$ . Find the probability that the sum of the number of white balls in box $A$ and the number of red balls in box $B$ is not a prime number. Round to four decimal places.
False propositions about events ●●○○○ 20 ptsConsider the following five propositions. ① Toss a coin twice. Let event $A$ be "both tosses land heads" and event $B$ be "both tosses land tails"; then $A$ and $B$ are complementary events. ② In proposition ①, $A$ and $B$ are mutually exclusive events. ③ Among 10 products, 3 are defective. Three products are selected at random. Let event $A$ be "at most 2 of the selected products are defective" and event $B$ be "at least 2 of the selected products are defective"; then $A$ and $B$ are mutually exclusive events. ④ If events $A$ and $B$ satisfy $P(A)+P(B)=1$ , then $A$ and $B$ are complementary events. ⑤ If $A$ and $B$ are mutually exclusive events, then $\overline{A}\cup\overline{B}$ is a certain event. Identify all the false propositions among these five and give their numbers, in increasing order.
Each player gets one ace ●●●○○ 50 ptsA deck of 52 cards containing four aces is shuffled thoroughly and the cards are dealt among four players so that each player receives 13 cards. Find the probability that each player receives exactly one ace. Give the answer as a percentage rounded to one decimal place.
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.
Draws to see all six unique balls ●●●○○ 50 ptsA jar contains 66 white balls and 6 unique balls: 1 red, 1 blue, 1 green, 1 purple, 1 yellow and 1 cyan. Balls are drawn one at a time, each ball is returned to the jar after it is drawn, and the colours are recorded. What is the expected number of draws needed until all 6 uniquely coloured balls have been seen at least once?
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.