Problem 44 - Algebra 2

Question

ACT/SAT For a game she's playing, Liz must draw a card from a deck of 26 cards, one with each letter of the alphabet on it, and roll a six-sided die. What is the probability that Liz will roll an odd number and draw a letter in her name? $$ \begin{array}{llll}{A} & {\frac{2}{3}} & {B} & {\frac{1}{13}} & {C} & {\frac{1}{26}} & {D} & {\frac{3}{52}}\end{array} $$

Step-by-Step Solution

Answer

The probability is \( \frac{3}{52} \).

Step 1: Determine Favorable Outcomes for the Die Roll

A six-sided die has numbers 1 through 6. For Liz to roll an odd number, the possible outcomes are 1, 3, and 5. That gives us 3 favorable outcomes for rolling an odd number.

Step 2: Determine Favorable Outcomes for Drawing a Card

Liz needs to draw a card with a letter from her name. Assuming Liz's name is 'LIZ', there are 3 letters: L, I, and Z. Therefore, there are 3 cards with these letters that represent favorable outcomes.

Step 3: Calculate Total Possible Outcomes

Since there are 26 cards in the deck, Liz has 26 possible outcomes for drawing a card. Additionally, with a six-sided die, there are 6 possible outcomes. The total possible outcomes when drawing a card and rolling a die is the product of these two: 26 cards times 6 die rolls equals 156 total outcomes.

Step 4: Calculate the Number of Joint Favorable Outcomes

The number of favorable outcomes for rolling an odd number is 3. The number of favorable outcomes for drawing a letter in her name is also 3. Therefore, the joint favorable outcomes (for both events happening) is 3 (odd numbers) times 3 (letters in her name), which totals 9 favorable outcomes.

Step 5: Calculate the Probability

Probability is calculated as the ratio of favorable outcomes to the total number of possible outcomes. Therefore, the probability that Liz will roll an odd number and draw a letter from her name is calculated by dividing the number of joint favorable outcomes by the total possible outcomes: \( \frac{9}{156} \). Simplifying this fraction gives us \( \frac{3}{52} \).

Key Concepts

Favorable Outcomes

In probability, identifying favorable outcomes is a key step in solving problems. It refers to the specific outcomes that fulfill the condition or event we are trying to accomplish. For any given problem, you'll first need to define what successful results look like. In the context of Liz's game, favorable outcomes for the die roll are when she gets an odd number. Since a six-sided die includes the numbers 1, 2, 3, 4, 5, and 6, favorable outcomes for rolling an odd number are:

• 1
• 3
• 5

The dice provides 3 favorable outcomes for odd numbers.
Similarly, when Liz draws a card, we look for cards with letters in her name. If her name is 'LIZ', the letters L, I, and Z are what make up favorable outcomes for the card draw. This also results in 3 cards being favorable. Recognizing these favorable outcomes sets the stage for calculating probability later on.

By understanding and identifying what constitutes a favorable outcome, you can effectively align your calculations towards assessing the likelihood of these particular scenarios happening.

Total Possible Outcomes

To calculate the probability of an event, knowing the total possible outcomes is crucial. This number considers all potential results from an action or a set of actions. For example, when Liz draws a card from a full deck of 26 cards and rolls a die, we calculate this as two separate actions.

• There are 26 possible outcomes from the card draw (one for every letter).
• There are 6 possible outcomes from rolling the die (one for each side).

Combining these two gives us the total possible outcomes for both events occurring together. Mathematically, this is achieved by multiplying the number of outcomes from each action, which results in 156 possible outcomes (26 cards multiplied by 6 die faces).

Understanding this total concept helps in parallelizing real-world actions like Liz's game, where multiple independent actions determine the final result. Without identifying total possible outcomes, assigning a probability correctly would not be feasible.

Simplifying Fractions

After determining both the favorable outcomes and the total possible outcomes, calculating probability involves setting these as a fraction. In Liz’s game, we found 9 joint favorable outcomes and 156 total possible outcomes. Therefore, the initial probability is represented as the fraction \( \frac{9}{156} \).
Simplifying fractions is the next step. This means reducing the fraction to its simplest form where the numerator and denominator no longer share any common factors other than 1.

• You need to find the greatest common divisor (GCD) of the numerator and the denominator.
• In this case, the GCD of 9 and 156 is 3.

By dividing both the numerator and the denominator by this GCD, the fraction \( \frac{9}{156} \) becomes \( \frac{3}{52} \).

This final result is or the simplest form of the probability that Liz will roll an odd number and draw a card that has a letter from her name. Simplifying fractions is important as it provides clarity and precision in answering probability questions, making results easier to interpret and understand.