Probability - Relative and Expected Frequencies

A biased coin is flipped $200$ times, and it lands on heads $124$ times. Calculate the relative frequency of the coin landing on heads.

$Relative\ Frequency = \frac{124}{200}$ $Relative\ Frequency = 0.62$

The relative frequency is found by taking the frequency of the specific outcome ($124$) and dividing it by the total number of experiments ($200$).

The probability that a computer component is faulty is $0.02$. In a batch of $5000$ components, how many would you expect to be faulty?

$Expected\ Frequency = n \times P(\text{faulty})$ $Expected\ Frequency = 5000 \times 0.02$ $Expected\ Frequency = 100$

To find the expected frequency, multiply the total number of trials ($n = 5000$) by the probability of the event ($P = 0.02$).

A bag contains red, blue, and yellow marbles. The relative frequency of picking a red marble is $0.4$ and a blue marble is $0.25$. If a marble is picked $80$ times, how many times would you expect to pick a yellow marble?

$P(\text{yellow}) = 1 - (0.4 + 0.25)$ $P(\text{yellow}) = 1 - 0.65 = 0.35$ $Expected\ Frequency = 80 \times 0.35 = 28$

First, find the probability of the missing outcome by subtracting the known relative frequencies from $1$. Then, multiply this probability by the total number of trials to find the expected frequency.