Question

a packet contains 20 lollies of which 6 are grape, 6 are blueberry, 8 are raspberry. if a lolly is drawn at random, find the probability that it is neither blueberry nor grape.

Explanation:

Step1: Find number of non - blueberry and non - grape lollies

The total number of lollies is 20. The number of blueberry lollies is 6 and the number of grape lollies is 6. So the number of lollies that are neither blueberry nor grape is the number of raspberry lollies, which is 8. Alternatively, we can calculate it as \(20-(6 + 6)=20 - 12=8\).

Step2: Calculate the probability

The probability of an event is given by the formula \(P=\frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}}\). Here, the number of favorable outcomes (neither blueberry nor grape) is 8 and the total number of outcomes is 20. So \(P=\frac{8}{20}\). Simplifying \(\frac{8}{20}\) by dividing both the numerator and the denominator by 4, we get \(\frac{8\div4}{20\div4}=\frac{2}{5}\)? Wait, no, wait. Wait, the number of raspberry lollies is 8. Wait, \(8/20=\frac{2}{5}\)? Wait, no, 8 divided by 20: 8 and 20 are both divisible by 4, 8÷4 = 2, 20÷4 = 5, so 2/5? Wait, but let's re - check. The number of lollies that are neither blueberry nor grape: total lollies 20, blueberry 6, grape 6. So non - blueberry and non - grape is 20-(6 + 6)=8. So probability is 8/20=\(\frac{8}{20}=\frac{2}{5}\)? Wait, no, 8/20 reduces to 2/5? Wait, 8 divided by 20: 8÷20 = 0.4, and 2/5 is 0.4. Alternatively, the number of lollies that are blueberry or grape is 6 + 6 = 12. So the number of lollies that are neither is 20 - 12 = 8. So probability is 8/20=\(\frac{2}{5}\). Wait, but let's check the options. One of the options is 2/5. Wait, but let's do it again.

Wait, the formula for probability of neither A nor B is \(P(\text{neither }A\text{ nor }B)=1 - P(A\cup B)\). \(P(A\cup B)=P(A)+P(B)-P(A\cap B)\). Since a lolly can't be both blueberry and grape (assuming), \(P(A\cap B) = 0\). So \(P(A\cup B)=\frac{6}{20}+\frac{6}{20}=\frac{12}{20}=\frac{3}{5}\). Then \(P(\text{neither }A\text{ nor }B)=1-\frac{3}{5}=\frac{2}{5}\). Yes, that's correct.

Answer:

\(\frac{2}{5}\) (corresponding to the option with \(\frac{2}{5}\))