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 3 points on the first draw, 10 points on the second draw, and 5 points on the third draw. the probability of selecting a letter worth 3 points on the first draw, 10 points on the second draw, and 5 points on the third draw is (type an integer or a simplified fraction.)

Explanation:

Step1: Count 3 - point letters

The 3 - point letters are B, C, M, P with a total of $2 + 2+2 + 2=8$ tiles. The probability of selecting a 3 - point letter on the first draw is $P_1=\frac{8}{100}$.

Step2: Count 10 - point letters

The 10 - point letters are J, Q, X, Z with a total of $1+1 + 1+1 = 4$ tiles. Since one tile is already drawn in the first step, there are 99 tiles left. So the probability of selecting a 10 - point letter on the second draw is $P_2=\frac{4}{99}$.

Step3: Count 5 - point letters

The 5 - point letters are K, V, W, Y with a total of $1+2 + 2+2=7$ tiles. Since two tiles are already drawn, there are 98 tiles left. So the probability of selecting a 5 - point letter on the third draw is $P_3=\frac{7}{98}$.

Step4: Calculate the combined probability

Since these are independent - draw events, the overall probability $P = P_1\times P_2\times P_3$.
\[

$$\begin{align*} P&=\frac{8}{100}\times\frac{4}{99}\times\frac{7}{98}\\ &=\frac{8\times4\times7}{100\times99\times98}\\ &=\frac{224}{970200}\\ &=\frac{8}{34650}\\ &=\frac{4}{17325} \end{align*}$$

\]

Answer:

$\frac{4}{17325}$