Basic Probability - ISEE Middle Level Quantitative Reasoning

1. What is $P(E)$ if $P(E^c)=\frac{3}{8}$?

   $\frac{5}{8}$. Subtract the probability of the complement from 1 to find the event's probability.

2. What is the probability of rolling an even number on a fair six-sided die?

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

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 in a standard deck.

4. A bag has $3$ red and $2$ blue marbles. What is $P(\text{red})$ on one draw?

   $\frac{3}{5}$. There are 3 red marbles out of 5 total marbles in the bag.

5. If events $A$ and $B$ are mutually exclusive, what is $P(A\cap B)$?

   $0$. Mutually exclusive events cannot occur together, so their intersection has zero probability.

6. What is the probability of an event that is impossible (has $0$ favorable outcomes)?

   $0$. An impossible event has no favorable outcomes, resulting in a probability of zero.

7. What is the probability of getting heads on a fair coin flip?

   $\frac{1}{2}$. One favorable outcome (heads) out of two equally likely outcomes on a fair coin.

8. What is the general addition rule for any events $A$ and $B$ (not necessarily disjoint)?

   $P(A\cup B)=P(A)+P(B)-P(A\cap B)$. The general rule accounts for overlap by subtracting the intersection's probability.

9. What is the probability of getting two heads in two fair coin flips?

   $\frac{1}{4}$. The probability of heads on each independent flip is $\frac{1}{2}$, so multiply for both.

10. What is the probability of an event $E$ if it has $f$ favorable outcomes out of $n$ equally likely outcomes?

    $P(E)=\frac{f}{n}$. Probability is defined as the ratio of the number of favorable outcomes to the total number of equally likely outcomes.

11. A bag has $3$ red and $2$ blue marbles. Without replacement, what is $P(\text{red then red})$?

    $\frac{3}{10}$. Without replacement, multiply $\frac{3}{5}$ by $\frac{2}{4}$ for two red draws.

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

    $P(A\cup B)=P(A)+P(B)$. For mutually exclusive events, the union's probability is the sum since there is no overlap.

13. What is the probability of an event that is certain (always occurs)?

    $1$. A certain event includes all possible outcomes, so its probability equals one.

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

    $\frac{1}{13}$. There are 4 aces out of 52 cards in a standard deck.

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

    $P(A\cap B)=P(A)P(B)$. For independent events, the joint probability is the product of individual probabilities.

16. 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 in a standard deck.

17. A bag has $3$ red and $2$ blue marbles. With replacement, what is $P(\text{red then red})$?

    $\frac{9}{25}$. With replacement, the probability remains $\frac{3}{5}$ for each independent draw, so multiply.

18. What is $P(E^c)$ if $P(E)=0.27$?

    $0.73$. The complement's probability is 1 minus the event's probability.

19. If $P(A)=\frac{1}{5}$ and $P(B)=\frac{2}{5}$ and $A,B$ are mutually exclusive, what is $P(A\cup B)$?

    $\frac{3}{5}$. Add the probabilities since the events are mutually exclusive with no overlap.

20. If $P(A)=\frac{3}{10}$ and $P(B)=\frac{1}{2}$ and $A,B$ are independent, what is $P(A\cap B)$?

    $\frac{3}{20}$. Multiply the probabilities since the events are independent.

21. What is the probability of rolling a $4$ on a fair six-sided die?

    $\frac{1}{6}$. One favorable outcome (rolling a 4) out of six equally likely outcomes on a fair die.

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

    $\frac{3}{13}$. There are 12 face cards (3 per suit, 4 suits) out of 52 in a standard deck.

23. What is the relationship between an event $E$ and its complement $E^c$?

    $P(E^c)=1-P(E)$. The probability of the complement is one minus the event's probability, as they together cover all outcomes.

24. What is the probability of getting at least one head in two fair coin flips?

    $\frac{3}{4}$. Calculated as 1 minus the probability of no heads (both tails), which is $\frac{1}{4}$.

25. If $P(A)=0.6$, $P(B)=0.5$, and $P(A\cap B)=0.2$, what is $P(A\cup B)$?

    $0.9$. Apply the general addition rule: add probabilities and subtract the intersection.

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