Question

the theoretical probability of drawing a red marble from a bag is $\frac{2}{5}$. the bag contains 60 marbles.
a. how many red marbles are in the bag?
b. a marble is drawn from the bag and replaced 80 times. how many times do you expect a red marble to be drawn?

Explanation:

Step1: Calculate the number of red marbles

The formula for probability is \(P=\frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}}\). Given \(P = \frac{2}{5}\) and total number of marbles \(n = 60\). Let the number of red marbles be \(x\). Then \(\frac{x}{60}=\frac{2}{5}\). Cross - multiply: \(5x=2\times60\).

Step2: Solve for \(x\)

\(5x = 120\), so \(x=\frac{120}{5}=24\).

Step3: Calculate the expected number of red marbles drawn

The formula for expected value \(E\) (in a binomial - like situation with replacement, where probability of success \(p=\frac{2}{5}\) and number of trials \(n = 80\)) is \(E=np\). Substitute \(n = 80\) and \(p=\frac{2}{5}\): \(E=80\times\frac{2}{5}\).

Step4: Simplify the expression for \(E\)

\(80\times\frac{2}{5}=\frac{80\times2}{5}=16\times2 = 32\).

Answer:

a. \(24\) red marbles.
b. \(32\) times.