- Compute probabilities of equally likely outcomes.
- Compute probabilities of the union of two events.
- Use the complement rule to find probabilities.
- Compute probability using counting theory.
Fantasy Quest Dice Game
You’re playing a fantasy quest game where players roll two six-sided dice to determine the success of their character’s actions. Different total values on the dice lead to different outcomes in the game.
Game Rules:
- Rolling a sum of 7 or 11 means Critical Success (your character performs an amazing feat)
- Rolling a sum of 2, 3, or 4 means Failure (your action doesn’t work)
- Rolling a sum of 10, 11, or 12 means High Roll (you get bonus points)
- Any other sum means a Standard Success
What is the probability of rolling a sum of 7 on two six-sided dice?
[latex]First, identify the sample space. When rolling two dice, there are [latex]6 \times 6 = 36[/latex] equally likely outcomes. Next, count the outcomes where the sum equals 7: (1,6) (2,5), (3,4), (4,3), (5,2), (6,1)[/latex] There are 6 ways to roll a sum of 7. [latex]P(\text{sum of 7}) = \dfrac{\text{number of favorable outcomes}}{\text{total number of outcomes}} = \dfrac{6}{36} = \dfrac{1}{6}[/latex] The probability of rolling a sum of 7 is [latex]\dfrac{1}{6}[/latex] or approximately 16.67%.[/latex]
In an advanced version of the game, you draw 3 cards from a special deck containing 5 Spell cards, 4 Weapon cards, and 3 Shield cards. What is the probability that you draw exactly 2 Spell cards and 1 Weapon card?