One genetic disease was tested positive in both parents of one family. It has been known that any child in this family has a 25% risk of inheriting this disease. A family has three children. The probability of this family having at least two children who inherited this genetic disease is:
To calculate the probability of this family having at least two children who inherited the genetic disease, we need to consider the different scenarios.
Let's use the following notation:
A: Child inherited the genetic disease
N: Child did not inherit the genetic disease
First, let's calculate the probability that two children have inherited the disease and one did not.
Probability of two children having the disease and one not having it:
P(AAN) = P(A) * P(A) * P(N) = 0.25 * 0.25 * 0.75 = 0.046875
Since there are three possible orders in which these children could be born, we multiply the above probability by 3:
P(at least two children have the disease) = 3 * 0.046875 = 0.140625
Next, let's calculate the probability that all three children have inherited the disease:
Probability of all three children having the disease:
P(AAA) = P(A) * P(A) * P(A) = 0.25 * 0.25 * 0.25 = 0.015625
Finally, let's calculate the probability that exactly two children have the disease:
Probability of two children having the disease:
P(AAA) = P(A) * P(A) * P(N) + P(A) * P(N) * P(A) + P(N) * P(A) * P(A) = 0.25 * 0.25 * 0.75 + 0.25 * 0.75 * 0.25 + 0.75 * 0.25 * 0.25 = 0.140625
To find the probability of this family having at least two children who inherited the genetic disease, we add up the probabilities of these three cases:
P(at least two children have the disease) + P(all three children have the disease) + P(exactly two children have the disease)
= 0.140625 + 0.015625 + 0.140625
= 0.296875
Therefore, the probability of this family having at least two children who inherited the genetic disease is approximately 0.296875 or 29.7%.
To calculate the probability of the family having at least two children who inherited the genetic disease, we need to consider the possible outcomes.
Since both parents have tested positive for the disease, they each carry one copy of the disease-causing gene. When they have children, each child has a 50% chance of inheriting the disease-causing gene from each parent.
Let's look at the possible scenarios for the three children:
1. All three children do not inherit the disease: The probability of a child not inheriting the disease is 75% (1 - 25%). The probability of this scenario is 0.75 * 0.75 * 0.75 = 0.4219 (or 42.19%).
2. One child inherits the disease, and the other two do not: The probability of one child inheriting the disease is 25% (or 0.25), while the probability of the other two children not inheriting the disease is 75% each. The probability of this scenario is 0.25 * 0.75 * 0.75 + 0.75 * 0.25 * 0.75 + 0.75 * 0.75 * 0.25 = 0.4219 (or 42.19%).
3. Two or more children inherit the disease: The probability of two or more children inheriting the disease is 0.25 * 0.25 * 0.75 + 0.25 * 0.75 * 0.25 + 0.75 * 0.25 * 0.25 + 0.25 * 0.25 * 0.25 = 0.1406 (or 14.06%).
To calculate the probability of the family having at least two children who inherited the disease, we sum up the probabilities of scenarios 2 and 3:
0.4219 + 0.1406 = 0.5625 (or 56.25%).
Therefore, the probability of this family having at least two children who inherited this genetic disease is 56.25%.