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4 problems match “symmetry argument”. Clear

Ace of spades before face cards ●●○○○ 20 ptsCarson flips over the cards of a standard 52-card deck one at a time. Find the probability that he flips over the ace of spades before any face card (jack, queen or king). Round to two decimal places.
First, fifteenth or last ball red ●●○○○ 20 ptsA box contains 100 balls, of which 40 are red. The balls are drawn from the box one at a time at random, without replacement. Determine: 1. the probability that the first ball drawn is red, 2. the probability that the fifteenth ball drawn is red, 3. the probability that the last ball drawn is red. Give each answer as a decimal.
Product after sum-replacement ●●●○○ 50 ptsOn a board there are 2023 distinct real numbers. Each number is simultaneously replaced by the sum of all the other numbers. After this replacement, the 2023 numbers on the board are the same as at the beginning (as a set of numbers). What value can the product of all the numbers on the board take?
Unique solution with a parameter ●●●○○ 50 ptsFind all values of the parameter $a$ for which the equation $a x^{2}+\sin ^{2} x=a^{2}-a$ has a unique solution. Give the value of $a$ .