Suppose that the proportions of blood phenotypes in a particular population are a follows:
A = .42
B = .10
AB = .04
O = .44
Assuming that the phenotypes of two randomly selected indiviuals are independent of one another, what is the probablility that both phenotypes are O? What is the probablility that the phenotypes of two randomly selected individuals match?
I'm really stuck on how to approach and solve this problem. From what I see, it says they are independent so would we need to use a formula of the kind
P(O Phenotype) = P(A) * P(B)
where A and B are the events of two randomly selected individuals?
Right!
now i don't know what to do frrom there
To solve this problem, you are correct that you need to use the concept of independent events. The probability that both phenotypes are O can be calculated by multiplying the probability of the first individual having phenotype O with the probability of the second individual also having phenotype O.
Let's first calculate the probability that the first individual has phenotype O. According to the given proportions, the probability of the first individual having phenotype O is 0.44.
Now, since the phenotypes of the two individuals are independent, the probability of the second individual also having phenotype O is also 0.44.
To find the probability that both individuals have phenotype O, you need to multiply these probabilities:
P(both phenotypes are O) = P(first individual has O) * P(second individual has O)
= 0.44 * 0.44
= 0.1936
Therefore, the probability that both phenotypes are O is 0.1936.
Next, let's find the probability that the phenotypes of two randomly selected individuals match. There are several possibilities for a match: both individuals can have phenotype A, both can have phenotype B, both can have phenotype AB, or both can have phenotype O.
So the probability of a match is the sum of the probabilities of these four scenarios:
P(match) = P(both individuals have type A) + P(both individuals have type B) + P(both individuals have type AB) + P(both individuals have type O)
Using the given proportions, we can calculate these probabilities:
P(both individuals have type A) = P(first individual has A) * P(second individual has A) = 0.42 * 0.42
P(both individuals have type B) = P(first individual has B) * P(second individual has B) = 0.10 * 0.10
P(both individuals have type AB) = P(first individual has AB) * P(second individual has AB) = 0.04 * 0.04
P(both individuals have type O) = P(first individual has O) * P(second individual has O) = 0.44 * 0.44
Calculating these probabilities:
P(both individuals have type A) = 0.1764
P(both individuals have type B) = 0.01
P(both individuals have type AB) = 0.0016
P(both individuals have type O) = 0.1936
Now, you can sum up these probabilities to find the probability of a match:
P(match) = P(both individuals have type A) + P(both individuals have type B) + P(both individuals have type AB) + P(both individuals have type O)
= 0.1764 + 0.01 + 0.0016 + 0.1936
= 0.3816
Therefore, the probability that the phenotypes of two randomly selected individuals match is 0.3816.
To solve this problem, you're on the right track in thinking about using the multiplication rule for independent events. Let's break down the problem step by step.
1. Probability that both phenotypes are O:
Since the phenotypes are independent, you can simply multiply the probabilities of each individual having phenotype O. Given that the proportion of phenotype O is 0.44, the probability of an individual having phenotype O is also 0.44. Therefore, the probability that both individuals have phenotype O is:
P(O phenotype for individual 1) * P(O phenotype for individual 2) = 0.44 * 0.44 = 0.1936 or 19.36%.
2. Probability that the phenotypes of two randomly selected individuals match:
Here, we need to consider all possible matching phenotypes: both being A, both being B, both being AB, or both being O. Let's calculate the probability for each case:
a. Probability of both being A:
P(A phenotype for individual 1) * P(A phenotype for individual 2) = 0.42 * 0.42 = 0.1764.
b. Probability of both being B:
P(B phenotype for individual 1) * P(B phenotype for individual 2) = 0.10 * 0.10 = 0.01.
c. Probability of both being AB:
P(AB phenotype for individual 1) * P(AB phenotype for individual 2) = 0.04 * 0.04 = 0.0016.
d. Probability of both being O:
P(O phenotype for individual 1) * P(O phenotype for individual 2) = 0.44 * 0.44 = 0.1936.
To find the probability that the phenotypes match, you sum up the probabilities for all the cases:
0.1764 + 0.01 + 0.0016 + 0.1936 = 0.3816 or 38.16%.
Therefore, the probability that the phenotypes of two randomly selected individuals match is 0.3816 or 38.16%.
Remember to always be careful with calculations involving probabilities, and don't forget to double-check your work!