Question

1. the probability that a student is late to school on any given day is 32%. find the probability that this student is late every day this week.
2. a high school surveys 400 students about their favorite elective and whether or not they play after school sports. among the surveyed students, 160 chose art as their favorite elective; 100 chose art and play after school sports. we know that students art and sports choices are independent of each other. how many students plan to participate in sports?
3. you are given a standard deck of 52 cards. three cards are chosen at random with replacement. what is the probability of choosing an ace, a spade, and a four?
4. dr. z has two bags of mixed candies. she lets a student pick one candy from each bag. the probability that the student picks a lollipop from the first bag is 1/6 and the probability that the student picks a lollipop from both bags is 1/15. what is the probability that she picks a lollipop from the second bag?
5. you roll two fair 6 - sided dice and add the numbers. what is the probability you roll a sum that is 9 or more?

Explanation:

Step1: Determine total possible outcomes

When rolling two 6 - sided dice, the total number of outcomes is \( 6\times6 = 36 \) (by the fundamental counting principle).

Step2: Find favorable outcomes

The pairs \((x,y)\) that give a sum \(S=x + y\geq9\):

• For \(S = 9\): \((3,6)\), \((4,5)\), \((5,4)\), \((6,3)\) (4 outcomes).
• For \(S = 10\): \((4,6)\), \((5,5)\), \((6,4)\) (3 outcomes).
• For \(S = 11\): \((5,6)\), \((6,5)\) (2 outcomes).
• For \(S = 12\): \((6,6)\) (1 outcome).

The total number of favorable outcomes is \(4 + 3+2 + 1=10\).

Step3: Calculate probability

The probability \(P=\frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}}=\frac{10}{36}=\frac{5}{18}\approx0.278\) (or \(27.8\%\) when converted to percentage).

Answer:

\( \frac{5}{18} \)