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6 problems match “complementary probability”. Clear

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1. Birthday not in December ●○○○○ 10 ptsIgnoring leap years, what is the probability that a randomly chosen person's birthday is not in December? Give your answer as a decimal rounded to three decimal places.
2. 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.
3. Probability of A from two unions ●●○○○ 20 ptsGiven $P(A \cup B)=0.7$ and $P(A \cup B')=0.9$, determine $P(A)$.
4. Birthday match among three people ●●○○○ 20 ptsThree people are in a room. Assuming each birthday is equally likely among 365 days (ignore leap years), find the probability that at least two of them share the same birthday. Round your answer to 9 decimal places.
5. 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.
6. 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%?