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 2 points on the first draw, 8 points on the second draw, and 4 points on the third draw. the probability of selecting a letter worth 2 points on the first draw, 8 points on the second draw, and 4 points on the third draw is (type an integer or a simplified fraction).

Explanation:

Step1: Count tiles for each point - value

From the Scrabble - tile distribution, letters worth 2 points are B, C, M, P (total 2 + 2+2 + 2=8 tiles), letters worth 8 points are J, X (total 1 + 1 = 2 tiles), and letters worth 4 points are F, H, V, W, Y (total 2+2 + 2+2 + 2 = 10 tiles).

Step2: Calculate the first - draw probability

The probability of selecting a 2 - point letter on the first draw is the number of 2 - point letters divided by the total number of tiles. So, $P_1=\frac{8}{100}$.

Step3: Calculate the second - draw probability

After the first draw, there are 99 tiles left. The probability of selecting an 8 - point letter on the second draw is $P_2=\frac{2}{99}$.

Step4: Calculate the third - draw probability

After the second draw, there are 98 tiles left. The probability of selecting a 4 - point letter on the third draw is $P_3=\frac{10}{98}$.

Step5: Calculate the combined probability

Since these are independent events (in the sense of sequential draws without replacement), the overall probability $P = P_1\times P_2\times P_3$.
\[

$$\begin{align*} P&=\frac{8}{100}\times\frac{2}{99}\times\frac{10}{98}\\ &=\frac{8\times2\times10}{100\times99\times98}\\ &=\frac{160}{970200}\\ &=\frac{4}{24255} \end{align*}$$

\]

Answer:

$\frac{4}{24255}$