a) You are playing a game with balls in a bag. The bag contains four different colors of balls: $5$ red, $3$ green, $4$ blue, and $10$ black. You will draw two balls from the bag without replacement, one at a time. You lose the game if exactly one of the balls is black (so if you draw two, you still win). What is the probability of winning this game? b) How many three letter words can you make out of the letters of the word MATHEMATICS? How many words can you make that are made entirely out of unique letters? How many words contain the letter T?

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Precalculus

Mean, variance, and standard deviation of binomial distributions

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Q. a) You are playing a game with balls in a bag. The bag contains four different colors of balls:

5

red,

3

green,

4

blue, and

10

black. You will draw two balls from the bag without replacement, one at a time. You lose the game if exactly one of the balls is black (so if you draw two, you still win). What is the probability of winning this game? b) How many three letter words can you make out of the letters of the word MATHEMATICS? How many words can you make that are made entirely out of unique letters? How many words contain the letter T?

Calculate Total Ways: To calculate the probability of winning the game, we first need to find the total number of ways to draw two balls from the bag. $\newline$ Total number of balls = $5$ red + $3$ green + $4$ blue + $10$ black = $22$ balls. $\newline$ Total ways to draw $2$ balls without replacement = $22$ choose $2$ = rac{22!}{2! imes (22 - 2)!} = $231$ .
Calculate Ways for Black Balls: Next, we calculate the number of ways to draw two black balls, since drawing two black balls is a winning outcome. $\newline$ Number of ways to draw $2$ black balls = $10$ choose $2$ = $\frac{10!}{2! \times (10 - 2)!}$ = $45$ .
Calculate Ways for Non-Black Balls: Now, we calculate the number of ways to draw two non-black balls, which is also a winning outcome. $\newline$ Number of non-black balls = $5$ red + $3$ green + $4$ blue = $12$ balls. $\newline$ Number of ways to draw $2$ non-black balls = $12$ choose $2$ = rac{12!}{2! * (12 - 2)!} = $66$ .
Calculate Total Winning Outcomes: The total number of winning outcomes is the sum of the ways to draw two black balls and the ways to draw two non-black balls. $\newline$ Total winning outcomes = Number of ways to draw $2$ black balls + Number of ways to draw $2$ non-black balls = $45 + 66 = 111$ .
Calculate Probability of Winning: The probability of winning the game is the number of winning outcomes divided by the total number of ways to draw two balls. $\newline$ Probability of winning = Total winning outcomes / Total ways to draw $2$ balls = $\frac{111}{231}$ .
Find Three-Letter Words: For the second part of the problem, we need to find the number of three-letter words that can be made from the word MATHEMATICS. $\newline$ The word MATHEMATICS has $11$ letters, but we have to consider the repeated letters: $M$ , $A$ , $T$ , and $I$ .
Find Three-Letter Words: For the second part of the problem, we need to find the number of three-letter words that can be made from the word MATHEMATICS. $\newline$ The word MATHEMATICS has $11$ letters, but we have to consider the repeated letters: M, A, T, and I.To find the number of three-letter words, we use permutations since the order matters. $\newline$ Number of three-letter words = $P(11, 3) = \frac{11!}{(11 - 3)!} = 11 \times 10 \times 9 = 990$ .