Question

a game of scrabble has 100 tiles, as shown to the right. the number of tiles for each letter are shown to the right of each tile. the letters point value is printed on the tile. three tiles are drawn in succession from scrabbles 100 tiles. find the probability of selecting a letter worth 4 points on the first draw, 10 points on the second draw, and 8 points on the third draw. the probability of selecting a letter worth 4 points on the first draw, 10 points on the second draw, and 8 points on the third draw is□ (type an integer or a simplified fraction.)

Explanation:

Step1: Determine the number of tiles for each point - value

• For 4 - point tiles: From the Scrabble tile distribution, the letters \( U\) (count = 4) and \( V\) (count = 2) have a 4 - point value. So, the total number of 4 - point tiles \(n_1=4 + 2=6\).
• For 10 - point tiles: The letters \( Q\) (count = 1) and \( Z\) (count = 1) have a 10 - point value. So, the total number of 10 - point tiles \(n_2 = 1+1=2\).
• For 8 - point tiles: The letters \( X\) (count = 1) have an 8 - point value. So, the total number of 8 - point tiles \(n_3=1\).

Step2: Calculate the probability of each draw

• The probability of the first draw (selecting a 4 - point tile): The total number of tiles is \(N = 100\). So, \(P_1=\frac{6}{100}\).
• After the first draw, there are \(N_1=99\) tiles left. The probability of the second draw (selecting a 10 - point tile) is \(P_2=\frac{2}{99}\).
• After the second draw, there are \(N_2 = 98\) tiles left. The probability of the third draw (selecting an 8 - point tile) is \(P_3=\frac{1}{98}\).

Step3: Calculate the combined probability

Since the draws are independent events (in the sense of conditional probability for dependent - without - replacement events), the combined probability \(P\) is the product of the individual probabilities.

$$P=\frac{6}{100}\times\frac{2}{99}\times\frac{1}{98}$$

$$P=\frac{6\times2\times1}{100\times99\times98}=\frac{12}{970200}=\frac{1}{80850}$$

Answer:

\(\frac{1}{80850}\)