Final answer:
The probability that a randomly selected fish from the lake weighs more than 12 kg, given a 60% chance of being species A and a 40% chance of being species B, is approximately 9.522%.
Step-by-step explanation:
To determine the probability that a randomly selected fish from this lake weighs more than 12 kg, we must take into account the combined probabilities associated with the two species of fish given their respective distributions.
Calculating Probability for Fish A
Species A fish are normally distributed with a mean weight of 10 kg and a standard deviation of 2 kg. Using the standard normal distribution, we calculate the z-score for a fish weighing more than 12 kg:
Z = (12 - 10) / 2 = 1
Consulting a z-table, or using a standard normal distribution calculator, we find the probability that a fish A weighs more than 12 kg corresponds to the right tail beyond z = 1. This gives us the probability of 0.1587.
Calculating Probability for Fish B
Species B fish have a normally distributed weight with a mean of 8 kg and a standard deviation of 1 kg. Calculation of the z-score for fish B weighing more than 12 kg is:
Z = (12 - 8) / 1 = 4
The probability of a fish B having a weight greater than 12kg is almost 0, as a z-score of 4 is extremely far in the tail of the standard normal distribution.
Combined Probability
Since there is a 60% chance of selecting species A fish and a 40% chance for species B, we multiply each species' probability of weighing more than 12 kg by these percentages and add them together to get the total probability.
Total Probability = (0.60 * 0.1587) + (0.40 * 0) = 0.09522 (approximately)
So, there is roughly a 9.522% chance that a randomly selected fish weighs more than 12 kg.