Real-World Probability - ISEE Lower Level Mathematics Achievement

1. What is $P(A)$ if $P(A^c)=0.27$?

   $0.73$. Apply the complement rule: subtract $P(A^c)$ from 1 to find $P(A)$.

2. What is the probability that a randomly chosen month has $31$ days?

   $\frac{7}{12}$. Seven months (Jan, Mar, May, Jul, Aug, Oct, Dec) have 31 days out of 12.

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

   $\frac{1}{2}$. Standard deck has 26 red cards out of 52 total cards.

4. 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 a fair die.

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

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

6. What is the probability of flipping two heads in $2$ fair coin flips?

   $\frac{1}{4}$. Independent flips each with probability $\frac{1}{2}$, so multiply for both heads.

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

   $\frac{1}{4}$. Thirteen hearts out of 52 cards in a standard deck.

8. What is the probability of choosing a green marble from $3$ blue and $5$ green marbles?

   $\frac{5}{8}$. Five green marbles out of eight total marbles.

9. What is the formula for probability when outcomes are equally likely?

   $P(A)=\frac{\text{favorable outcomes}}{\text{total outcomes}}$. When all outcomes are equally likely, probability is the ratio of favorable to total outcomes.

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

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

11. What inequality must any probability $P(A)$ satisfy?

    $0\le P(A)\le 1$. Probabilities range from zero for impossibility to one for certainty, inclusive.

12. What is the probability of choosing a blue marble from $3$ blue and $5$ green marbles?

    $\frac{3}{8}$. Three blue marbles out of eight total marbles.

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

    $\frac{1}{13}$. Four aces in a 52-card deck simplify to this fraction.

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

    $\frac{1}{4}$. One vowel (A) among the four letters M, A, T, H.

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

    $P(A^c)=1-P(A)$. The complement covers all outcomes not in $A$, so its probability is one minus $P(A)$.

16. What is the probability of an event that is certain?

    $1$. A certain event includes all possible outcomes, yielding probability one.

17. What is the probability of flipping at least one head in $2$ fair coin flips?

    $\frac{3}{4}$. Out of four possible outcomes in two flips, three include at least one head.

18. What is the probability that a random integer from $1$ to $10$ is a multiple of $3$?

    $\frac{3}{10}$. Three multiples of 3 (3,6,9) from integers 1 to 10.

19. What is the probability of an event that is impossible?

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

20. What is $P(A\text{ or }B)$ if $A$ and $B$ are mutually exclusive?

    $P(A\text{ or }B)=P(A)+P(B)$. Mutually exclusive events have no overlap, so add their probabilities for the union.

21. What is the probability of not rolling a $6$ on one fair die?

    $\frac{5}{6}$. Complement of rolling a 6, which has probability $\frac{1}{6}$, subtracted from 1.

22. What is the probability that a randomly chosen day of the week is a weekend day?

    $\frac{2}{7}$. Two weekend days (Saturday, Sunday) out of seven days in a week.

23. What is $P(A\text{ and }B)$ if $A$ and $B$ are independent?

    $P(A\text{ and }B)=P(A)\cdot P(B)$. Independent events' occurrences don't affect each other, so multiply probabilities for intersection.

24. What is the probability that a random integer from $1$ to $20$ is prime?

    $\frac{8}{20}$. Eight primes (2,3,5,7,11,13,17,19) from integers 1 to 20.

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