This is an unbiased spinner with ten numbered sectors. Four of the sectors are shaded. When spun, the probability of stopping at any one of the sectors is equal.
(a) If the pointer is spun once, find the probability that the number is a prime number OR the sector is shaded
My answer:
$$P = 5/10 + 4/10 = 9/10$$
(b) If the pointer is spun once, find probability that the number is even AND the sector is shaded
My answer:
$$P= 4/10 \times 4/10 = 4/25$$
However, both of my answers are wrong and I don't quite understand why. Can I get help? Thanks in advance!
Best Answer
For this problem, you cannot use the addition rule because the events are not mutually exclusive: that is, it is possible that a sector is shaded AND the number in it is prime. (An example of mutually exclusive events would be, either it is raining or it is not raining, it cannot be both at the same time.) You cannot use the multiplication rule either as the conditions mentioned are not independent; e.g., the probability of being shaded is different for primes, odd numbers, etc. So, it is better to count them by hand.
For part (a), there are 8 sectors which either are shaded or contain a prime number. Thus, the probability of getting one of these is $8/10$.
For part (b), there is only one sector which is shaded and contains an even number. Thus, the probability is $1/10$.
(Extra: if you really want to use the addition rule, you can add the probability that a sector is shaded to the probability that a sector contains a prime number, and subtract the probability that both conditions occur together. You need to subtract because the case in which both occur is counted once in the first probability, and again in the second. This gives the same answer. If you really want to use the multiplication rule, well, you can't.)