Question

the game of scrabble has 100 tiles. the diagram shows the number of tiles for each letter and the letter’s point value. one tile is drawn from scrabble’s 100 tiles. if one tile is drawn from the 100 tiles, find the odds in favor and the odds against selecting a letter worth 1 point. the odds in favor of selecting a letter worth 1 point are \\(\square:\square\\) (simplify your answers.)

Explanation:

Step1: Count the number of favorable and unfavorable outcomes

From the diagram, the letters worth 1 - point are \(E(12), A(9), I(9), O(8), N(6), S(4), T(6), L(4), R(6), U(4), D(4), G(3), B(2), C(2), M(2), P(2), H(4), V(2), W(2), Y(2), F(2), K(1), J(1), X(1), Q(1), Z(1)\).
Counting them: \(12 + 9+9 + 8+6 + 4+6 + 4+6 + 4+4 + 3+2 + 2+2 + 2+4 + 2+2 + 2+2 + 1+1 + 1+1 + 1= 91\) (unfavorable).
The number of 1 - point tiles: \(91\). The number of non - 1 - point tiles: \(100 - 91=9\) (favorable, since we want to find odds in favor of 1 - point, actually there is a mistake in the problem setup, odds in favor of an event \(E\) is \(\frac{n(E)}{n(\text{not }E)}\), odds against is \(\frac{n(\text{not }E)}{n(E)}\). Assuming the problem means odds in favor of non - 1 - point (since if we consider the correct formula for odds in favor of an event \(E\): \(O(E)=\frac{n(E)}{n(\text{not }E)}\), and odds against \(O(\text{not }E)=\frac{n(\text{not }E)}{n(E)}\)). Let \(E\) be the event of selecting a non - 1 - point tile. \(n(E)=9\), \(n(\text{not }E)=91\)

Step2: Calculate the odds in favor and odds against

Odds in favor of selecting a non - 1 - point tile (if we assume the problem has a mis - labeling and we calculate based on the correct formula for odds):
Odds in favor \(=\frac{9}{91}\)
Odds against \(=\frac{91}{9}\)

If we assume the problem actually wants odds in favor of 1 - point (wrong formula application in terms of standard probability, but if we follow the problem's (maybe wrong) wording):
Let \(E\) be the event of selecting a 1 - point tile. \(n(E) = 91\), \(n(\text{not }E)=9\)
Odds in favor \(=\frac{91}{9}\)
Odds against \(=\frac{9}{91}\)

Answer:

Odds in favor: \(\frac{91}{9}\), Odds against: \(\frac{9}{91}\)