Colour ratios

Solve the first problem 18 problems

Blue balls with twice the red probability ●○○○○ 10 ptsA bag contains 6 red balls and some blue balls. The probability of drawing a blue ball from the bag is twice the probability of drawing a red ball. Find the number of blue balls in the bag.
Estimating red balls from draw frequencies ●○○○○ 10 ptsAn opaque bag contains a total of $20$ balls, each either red or white, identical except for color. A ball is drawn at random, its color is noted, and it is put back; this is repeated many times. The results are: - out of $50$ draws, $11$ red - out of $100$ draws, $27$ red - out of $200$ draws, $50$ red - out of $500$ draws, $124$ red - out of $800$ draws, $201$ red - out of $1000$ draws, $249$ red Estimate the number of red balls in the bag.
Guaranteeing a red ball in draws ●○○○○ 10 ptsA bag contains 10 red balls, 8 white balls and 7 black balls. The balls are drawn one by one at random, without replacement. Find the minimum value of $k$ such that a red ball is guaranteed to have been drawn by the $k$ -th draw (on the $k$ -th draw or earlier).
Probability the game never stops ●●○○○ 20 ptsA box starts with one red ball and one blue ball. At each step, a ball is taken at random from the box. If it is red, the game stops. If it is blue, it is returned to the box together with one additional blue ball. Thus, at the $n$ -th draw the box contains $n+1$ balls, one of which is red. Find the probability that the game never stops.
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.
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.
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 $\lfloor 2021 p \rfloor$ .
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?

More in Drawing from bags

Marbles in bags Number of balls from a probability Several draws Socks