Probability - Dice: You roll two fair six-sided dice. What is the probability that the product of the two numbers rolled is even?
Algebra - Quadratics: Solve the quadratic equation: x^2 - 5x + 6 = 0.
Statistics - Mean and Median: The ages of a group of people are: 18, 22, 25, 27, 28, 30, 35. What are the mean and median ages of the group?
Math - Sequences: Find the next term in the arithmetic sequence: 2, 5, 8, 11, …
Probability - Cards: You draw a card from a standard deck of 52 playing cards. What is the probability that the card is a red face card (king, queen, or jack)?
Algebra - Systems of Equations: Solve the system of equations: 2x - 3y = 4 and 5x + 4y = -1.
Statistics - Variability: The test scores of a class are: 70, 85, 90, 95, 80, 75. Calculate the range and standard deviation of the test scores.
Math - Geometry: The base of a triangle is 12 cm, and its height is 8 cm. What is the area of the triangle?
Probability - Conditional Probability: You draw two cards from a deck without replacement. Given that the first card drawn is a heart, what is the probability that the second card drawn is also a heart?
Math - Exponents: Simplify the expression: (3^4 * 3^3) / 3^5.
Probability - Dice: There are 36 possible outcomes when rolling two dice. Outcomes with an even product are those where at least one die shows an even number (2, 4, or 6). There are 3 even numbers on a die, so 3 out of 6 outcomes for each die result in an even product. The probability is (3/6) * (3/6) + (3/6) * (3/6) = 1/4 + 1/4 = 1/2.
Algebra - Quadratics: Factor the quadratic equation: x^2 - 5x + 6 = 0 (x - 2)(x - 3) = 0 Solve for x: x = 2, x = 3.
Statistics - Mean and Median: Mean = (18 + 22 + 25 + 27 + 28 + 30 + 35) / 7 ≈ 26.43 The median is the middle value: 27.
Math - Sequences: The common difference in the arithmetic sequence is 5 - 2 = 8 - 5 = 3. The next term is 11 + 3 = 14.
Probability - Cards: The deck has 26 red cards, of which 6 are red face cards (3 kings, 3 queens, and 3 jacks in hearts and diamonds). The probability is 6/52 = 3/26.
Algebra - Systems of Equations: Solve using substitution or elimination: 2x - 3y = 4 5x + 4y = -1 Multiply the first equation by 4 and add to the second equation: 8x - 12y = 16 5x + 4y = -1 13x - 8y = 15 13x = 15 + 8y x = (15 + 8y) / 13 Substitute into the first equation: 2((15 + 8y) / 13) - 3y = 4 30 + 16y - 39y = 52 16y - 39y = 22 -23y = 22 y = -22 / 23 x = (15 - 176) / 13 x = -161 / 13
Statistics - Variability: Range = 95 - 70 = 25 Mean = (70 + 85 + 90 + 95 + 80 + 75) / 6 = 82.5 Variance = [(70 - 82.5)^2 + … + (75 - 82.5)^2] / 6 ≈ 72.92 Standard deviation = √72.92 ≈ 8.54
Math - Geometry: Area of the triangle = (1/2) * base * height = (1/2) * 12 * 8 = 48 square cm.
Probability - Conditional Probability: Given that the first card drawn is a heart, there are 12 hearts left in the deck and 51 cards remaining. The probability that the second card is also a heart is 12/51.
Math - Exponents: Simplify the expression: (3^4 * 3^3) / 3^5 Combine exponents: 3^(4 + 3) / 3^5 Simplify: 3^7 / 3^5 Subtract exponents: 3^(5) implify: 3^2 Final answer: 9.