Calculating Probability - ISEE Middle Level Mathematics Achievement

1. What is the probability of choosing a vowel from the letters in $\text{MATH}$?

   $\frac{1}{4}$. The word MATH has four distinct letters, with one vowel (A) equally likely to be chosen.

2. What is the probability of choosing a multiple of $3$ from $\{1,2,3,4,5,6,7,8,9\}$?

   $\frac{1}{3}$. The set has nine numbers, with three multiples of 3 (3, 6, 9) equally likely to be chosen.

3. What is the probability of rolling a $5$ on a fair six-sided die?

   $\frac{1}{6}$. A fair die has six equally likely outcomes, with only one favorable for rolling a 5.

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

   $\frac{3}{4}$. Two coin flips yield four equally likely outcomes, with at least one head in three of them (HH, HT, TH).

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

   $\frac{1}{13}$. A standard deck has 52 cards, with 4 aces equally likely to be drawn.

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

   $\frac{1}{4}$. A standard deck has 52 cards, with 13 hearts equally likely to be drawn.

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

   $P(A\cap B)=P(A)\cdot P(B)$. For independent events, the joint probability multiplies individual probabilities as one does not affect the other.

8. What is $P(A^c)$ if $P(A)=\frac{2}{5}$?

   $\frac{3}{5}$. The complement rule gives $1 - \frac{2}{5} = \frac{3}{5}$ for the probability of not $A$.

9. What is the probability of a certain event?

   $1$. A certain event includes all possible outcomes, equaling the total probability of 1.

10. What is the probability of an impossible event?

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

11. What is the probability of rolling a number greater than $4$ on a fair die?

    $\frac{1}{3}$. Numbers greater than 4 (5, 6) are two of the six equally likely outcomes on a fair die.

12. What is $P(A\cup B)$ if $P(A)=\frac{1}{2}$, $P(B)=\frac{1}{3}$, and $P(A\cap B)=\frac{1}{6}$?

    $\frac{2}{3}$. Use the general addition rule: $\frac{1}{2} + \frac{1}{3} - \frac{1}{6} = \frac{3}{6} + \frac{2}{6} - \frac{1}{6} = \frac{4}{6}$.

13. What is the probability of choosing a prime number from $\{1,2,3,4,5,6,7,8,9,10\}$?

    $\frac{2}{5}$. The set has ten numbers, with four primes (2, 3, 5, 7) equally likely to be chosen.

14. What is the addition rule for any events $A$ and $B$ (may overlap)?

    $P(A\cup B)=P(A)+P(B)-P(A\cap B)$. The general addition rule subtracts the intersection to correct for overlapping outcomes in the union.

15. 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 probability is the sum since there is no overlap.

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

    $\frac{3}{13}$. A standard deck has 52 cards, with 12 face cards (J, Q, K in 4 suits) equally likely to be drawn.

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

    $\frac{1}{2}$. A fair coin has two equally likely outcomes, with heads as one of them.

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

    $\frac{1}{2}$. Even numbers (2, 4, 6) are three of the six equally likely outcomes on a fair die.

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

    $\frac{1}{4}$. Two coin flips yield four equally likely outcomes, with two heads (HH) in one of them.

20. What is the complement rule for an event $A$?

    $P(A^c)=1-P(A)$. The complement rule subtracts the probability of event $A$ from 1 to find the probability of not $A$.

21. What is $P(A\cap B)$ if $P(A)=\frac{3}{5}$ and $P(B)=\frac{2}{7}$ and events are independent?

    $\frac{6}{35}$. For independent events, multiply: $\frac{3}{5} \times \frac{2}{7} = \frac{6}{35}$.

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

    $\frac{7}{12}$. For mutually exclusive events, add probabilities: $\frac{1}{4} + \frac{1}{3} = \frac{3}{12} + \frac{4}{12} = \frac{7}{12}$.

23. What is the probability of getting exactly one head in two fair coin flips?

    $\frac{1}{2}$. Two coin flips yield four equally likely outcomes, with exactly one head in two of them (HT, TH).

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

    $\frac{1}{2}$. A standard deck has 52 cards, with 26 red cards equally likely to be drawn.

25. What is the probability formula when all outcomes are equally likely?

    $P(\text{event})=\frac{\text{favorable outcomes}}{\text{total outcomes}}$. This formula divides the number of favorable outcomes by the total number of equally likely outcomes in the sample space.

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