Single and Compound Probability - ISEE Upper Level Mathematics Achievement

1. What is the probability of rolling a $2$ or a $5$ on one fair die?

   $\frac{1}{3}$. Two favorable outcomes ($2$ or $5$) out of six, and events are mutually exclusive.

2. What is the general addition rule for any events $A$ and $B$?

   $P(A\cup B)=P(A)+P(B)-P(A\cap B)$. Accounts for overlap by subtracting the intersection probability from the sum of individual probabilities.

3. What is the probability of drawing a red card from a standard $52$-card deck?

   $\frac{1}{2}$. There are $26$ red cards (hearts and diamonds) out of $52$ total cards.

4. What is the complement rule for an event $A$ in probability notation?

   $P(A^c)=1-P(A)$. Calculates the probability of the complement by subtracting the event's probability from 1, as they are mutually exclusive and exhaustive.

5. What is the probability of drawing an ace from a standard $52$-card deck?

   $\frac{1}{13}$. Four aces (one per suit) out of $52$ cards in the deck.

6. What is $P(A\cap B)$ if $P(A)=\frac{2}{5}$ and $P(B\mid A)=\frac{3}{4}$?

   $\frac{3}{10}$. Multiplies $P(A)$ by conditional $P(B\mid A)$ for joint probability of dependent events.

7. What is the probability of rolling a $6$ on a fair six-sided die?

   $\frac{1}{6}$. One favorable outcome ($6$) out of six equally likely faces on the die.

8. What is the probability of drawing two aces in a row without replacement from a $52$-card deck?

   $\frac{4}{52}\cdot\frac{3}{51}=\frac{1}{221}$. Multiplies probabilities of drawing first ace then second without replacement.

9. What is the probability of rolling doubles with two fair six-sided dice?

   $\frac{1}{6}$. Six double outcomes out of $36$ total combinations for two dice.

10. What is the addition rule for mutually exclusive events $A$ and $B$?

    $P(A\cup B)=P(A)+P(B)$. Adds probabilities directly for mutually exclusive events since their intersection is empty.

11. What is the probability formula for an event $A$ using favorable and total outcomes?

    $P(A)=\frac{\text{favorable}}{\text{total}}$. Defines probability as the ratio of favorable outcomes to total possible outcomes in a sample space.

12. What is the probability of drawing a face card (J, Q, or K) from a $52$-card deck?

    $\frac{3}{13}$. Twelve face cards ($3$ per suit across $4$ suits) out of $52$ total cards.

13. What is $P(\text{not }6)$ when rolling one fair die?

    $\frac{5}{6}$. Uses the complement rule: $1$ minus the probability of rolling a $6$ ($\frac{1}{6}$).

14. What is the probability of flipping two fair coins and getting exactly one head?

    $\frac{1}{2}$. Two favorable outcomes (HT, TH) out of four possible results for two coins.

15. What is the probability of drawing two red cards in a row with replacement from a $52$-card deck?

    $\frac{1}{2}\cdot\frac{1}{2}=\frac{1}{4}$. Independent draws with replacement, each with $\frac{1}{2}$ probability of red.

16. What is the probability of rolling two fair dice and getting a sum of $7$?

    $\frac{1}{6}$. Six ways to get sum $7$ out of $36$ possible outcomes for two dice.

17. What is the probability of flipping two fair coins and getting two heads?

    $\frac{1}{4}$. Independent coin flips each with $\frac{1}{2}$ probability of heads, multiplied for both.

18. What is the conditional probability formula for $P(A\mid B)$?

    $P(A\mid B)=\frac{P(A\cap B)}{P(B)}$. Divides the joint probability by the probability of the conditioning event to find the likelihood given that event.

19. What is the probability of rolling an even number on a fair die?

    $\frac{1}{2}$. Three even numbers ($2,4,6$) out of six possible outcomes on the die.

20. What is the probability of flipping a fair coin and getting heads?

    $\frac{1}{2}$. Two equally likely outcomes (heads or tails) on a fair coin, with one favorable.

21. What is the multiplication rule for dependent events using conditional probability?

    $P(A\cap B)=P(A)P(B\mid A)$. Expresses joint probability using the initial event and the conditional probability for dependent events.

22. What is the probability of drawing a heart from a standard $52$-card deck?

    $\frac{1}{4}$. There are $13$ hearts out of $52$ cards, yielding the ratio of favorable to total outcomes.

23. What is the multiplication rule for independent events $A$ and $B$?

    $P(A\cap B)=P(A)P(B)$. Multiplies probabilities for independent events because one does not affect the other.

24. What is $P(A\cup B)$ if $P(A)=\frac{1}{3}$, $P(B)=\frac{1}{4}$, and $A,B$ are mutually exclusive?

    $\frac{7}{12}$. Adds probabilities since events are mutually exclusive, with no overlap.

25. What is the probability of drawing two hearts in a row without replacement from a $52$-card deck?

    $\frac{13}{52}\cdot\frac{12}{51}=\frac{1}{17}$. Multiplies conditional probabilities for first and second heart without replacement.

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