A shop stocks tinned cat food of two makes, A and B, and two sizes, large and small. Of the stock, 70%
is of brand A, 30% is of the brand B. Of the tins of brand A, 30% are of the small size, whilst of the tins
of brand B, 40% are of the small size. Determine the probability that
(a) a tin chosen at random from the stock will be of the small size.
(b) a small tin chosen at random from the stock will be brand A.
If the events are independent, the probability of both/all events occurring is determined by multiplying the probabilities of the individual events.
A. .3 * .4 = ?
B. .7 * .3 = ?
Let's calculate the probabilities step-by-step.
(a) To determine the probability that a tin chosen at random from the stock will be of the small size, we need to calculate the overall percentage of small-sized tins.
First, we calculate the probability of selecting a small-sized tin from brand A:
Probability of selecting a small-sized tin from brand A = 70% (of brand A) × 30% (of small size) = 0.7 × 0.3 = 0.21 or 21%
Next, we calculate the probability of selecting a small-sized tin from brand B:
Probability of selecting a small-sized tin from brand B = 30% (of brand B) × 40% (of small size) = 0.3 × 0.4 = 0.12 or 12%
Finally, we add these two probabilities together to get the overall probability of selecting a small-sized tin from the stock:
Overall probability of selecting a small-sized tin = 21% + 12% = 33% or 0.33
Therefore, the probability that a tin chosen at random from the stock will be of the small size is 33% or 0.33.
(b) To determine the probability that a small tin chosen at random from the stock will be brand A, we need to calculate the percentage of small-sized tins that are of brand A.
The probability of selecting a small-sized tin from brand A was calculated earlier as 21% or 0.21.
Now, to determine the probability that a small tin chosen at random from the stock will be brand A, we need to divide the percentage of small-sized tins from brand A by the overall percentage of small-sized tins:
Probability of selecting a small-sized tin from brand A / Overall probability of selecting a small-sized tin = 0.21 / 0.33 ≈ 0.636 or 63.6%
Therefore, the probability that a small tin chosen at random from the stock will be brand A is approximately 63.6%.
To determine the probability of a tin chosen at random from the stock being of the small size, we need to find the combined probability of selecting a small tin from brand A and from brand B, and then sum these probabilities.
(a) Probability of selecting a small tin from brand A:
We know that 70% of the stock is brand A, and of those, 30% are small tins. Therefore, the probability of selecting a small tin from brand A is: (70% * 30%) = 21%.
(b) Probability of selecting a small tin from brand B:
We know that 30% of the stock is brand B, and of those, 40% are small tins. Therefore, the probability of selecting a small tin from brand B is: (30% * 40%) = 12%.
Now, to determine the overall probability of selecting a small tin, we sum up the probabilities from brand A and brand B:
Probability of selecting a small tin = Probability of selecting a small tin from brand A + Probability of selecting a small tin from brand B = 21% + 12% = 33%.
Therefore, the probability that a tin chosen at random from the stock will be of the small size is 33%.
To determine the probability that a small tin chosen at random from the stock will be brand A, we need to find the probability of selecting a small tin from brand A relative to all small tins.
Probability of a small tin being brand A:
We know that 70% of the stock is brand A and of those, 30% are small tins. Therefore, we can calculate the probability of a small tin being brand A as (70% * 30%) = 21%.
Therefore, the probability that a small tin chosen at random from the stock will be brand A is 21%.