Question

1. what is the probability of the spinner? 2. what is the probability of tossing a number cube, with faces labeled 1 - 6, and spinning this spinner, and getting a sum of 9 or greater? 3. cadie is going to simultaneously. what is the probability that exactly two on heads? a. 1/6 b. 1/4 c. 1/3 d. 3/8 4. an ordinary deck of hearts, diamonds, clubs, spades. each suit has 13 different cards. there are 52 cards in all. one card is drawn at random from the deck. what is the probability that the card is a heart or clubs?

Explanation:

1.

Step1: Count total sections

The spinner has 8 equal - sized sections.

Step2: Count favorable sections

There are 3 sections labeled 'Y'.

Step3: Calculate probability

The probability \(P=\frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}}=\frac{3}{8}\)

Step1: Find total number of outcomes

The number cube has 6 outcomes and the spinner has 8 outcomes. So the total number of combined outcomes is \(6\times8 = 48\).

Step2: List pairs and find favorable outcomes

Let the number on the cube be \(n\) and the number on the spinner be \(s\). We want \(n + s\geq9\).
When \(n = 3\), \(s\) must be 6 (but no 6 on spinner, 0 favorable).
When \(n = 4\), \(s\) can be 5, 0 favorable.
When \(n=5\), \(s\) can be 4, 5, 8 favorable (since 4 and 5 are on spinner).
When \(n = 6\), \(s\) can be 3, 4, 5, 12 favorable.
The total number of favorable outcomes is \(8 + 12=20\).

Step3: Calculate probability

\(P=\frac{20}{48}=\frac{5}{12}\) (There seems to be an error in the provided options as \(\frac{5}{12}\) is not among them. But following the correct process).

3.

Step1: Use binomial probability formula

The binomial probability formula is \(P(X = k)=C(n,k)\times p^{k}\times(1 - p)^{n - k}\), where \(n\) is the number of trials, \(k\) is the number of successes, \(p\) is the probability of success on a single - trial, and \(C(n,k)=\frac{n!}{k!(n - k)!}\). Here, \(n = 3\) (number of coin - tosses), \(k = 2\) (number of heads), and \(p=\frac{1}{2}\) (probability of getting a head in a single coin - toss).

Step2: Calculate combination \(C(n,k)\)

\(C(3,2)=\frac{3!}{2!(3 - 2)!}=\frac{3!}{2!1!}=\frac{3\times2!}{2!×1}=3\).

Step3: Calculate probability

\(P(X = 2)=C(3,2)\times(\frac{1}{2})^{2}\times(1-\frac{1}{2})^{3 - 2}=3\times\frac{1}{4}\times\frac{1}{2}=\frac{3}{8}\)

Answer:

C. \(\frac{3}{8}\)

2.