Sickle cell anemia is an inherited blood disease that occurs...

Sickle cell anemia is an inherited blood disease that occurs primarily in blacks. In the United States, about 15 of every 10,000 black children have sickle cell anemia. The red blood cells of an affected person are abnormal; the result is severe chronic anemia (inability to carry the required amount of oxygen), which causes headaches, shortness of breath, jaundice, increased risk of pneumococcal pneumonia and gallstones, and other severe problems. Sickle cell anemia occurs in children who inherit an abnormal type of hemoglobin, called hemoglobin S, from both parents. If hemoglobin S is inherited from only one parent, the person is said to have sickle cell trait and is generally free from symptoms. There is a $50 \%$ chance that a person who has sickle cell trait will pass hemoglobin $\mathrm{S}$ to an offspring. a. Obtain the probability that a child of two people who have sickle cell trait will have sickle cell anemia. b. If two people who have sickle cell trait have five children, determine the probability that at least one of the children will have sickle cell anemia. c. If two people who have sickle cell trait have five children, find the probability distribution of the number of those children who will have sickle cell anemia. d. Construct a probability histogram for the probability distribution in part (c). e. If two people who have sickle cell trait have five children, how many can they expect will have sickle cell anemia?

Sickle cell anemia is an inherited blood disease that occurs primarily in blacks. In the United States, about 15 of every 10,000 black children have sickle cell anemia. The red blood cells of an affected person are abnormal; the result is severe chronic anemia (inability to carry the required amount of oxygen), which causes headaches, shortness of breath, jaundice, increased risk of pneumococcal pneumonia and gallstones, and other severe problems. Sickle cell anemia occurs in children who inherit an abnormal type of hemoglobin, called hemoglobin S, from both parents. If hemoglobin S is inherited from only one parent, the person is said to have sickle cell trait and is generally free from symptoms. There is a $50 \%$ chance that a person who has sickle cell trait will pass hemoglobin $\mathrm{S}$ to an offspring.
a. Obtain the probability that a child of two people who have sickle cell trait will have sickle cell anemia.
b. If two people who have sickle cell trait have five children, determine the probability that at least one of the children will have sickle cell anemia.
c. If two people who have sickle cell trait have five children, find the probability distribution of the number of those children who will have sickle cell anemia.
d. Construct a probability histogram for the probability distribution in part (c).
e. If two people who have sickle cell trait have five children, how many can they expect will have sickle cell anemia?

Key Concepts

Probability Histogram

A probability histogram is a graphical representation of a probability distribution where the heights of the bars correspond to the probabilities of different outcomes. In the binomial setting for genetic predictions, this helps in visualizing the likelihood of various numbers of offspring being affected by a particular genetic disorder.

Expected Value in Binomial Distribution

The expected value in a binomial distribution represents the average number of successes (or occurrences of a specific event) expected over a set of trials. It is calculated by multiplying the number of trials by the probability of success in each trial. This concept is used to predict the number of children likely to inherit a genetic condition when the number of children and individual probabilities are known.

Probability of Independent Events

The probability of independent events means that the outcome of one event does not affect the outcome of another. In genetics, if each parent has a 50% chance of passing a particular allele, the chance that a specific combination occurs in the offspring is found by multiplying the probabilities of the independent events. This is crucial for computing the likelihood of a child inheriting a particular genotype.

Mendelian Inheritance

Mendelian inheritance involves the principles of how traits or disorders are passed from parents to offspring through genes. It explains that when each parent carries one copy of an abnormal gene (heterozygous), the chance of their child inheriting the abnormal gene from both parents, and therefore expressing the disease, follows predictable ratios. This concept is fundamental in understanding genetic transmission and calculating the probability that a child will have a genetic disorder.

Binomial Distribution

The binomial distribution is used when there are a fixed number of independent trials, each with the same probability of success, and the interest lies in the number of successes. In the context of genetic events, when multiple children are considered, each child's chance of inheriting a certain trait is an independent trial, and the overall distribution of outcomes (e.g., number of children with the disorder) follows a binomial pattern.

Transcript

00:01 So we know that the probability that a trait is passed on by a parent that has it is 50%.

00:07 So what's the probability that two parents pass on and have a child that has sickle cell? so the probability that two parents pass on and you have a child who has the sickle cell.

00:22 And in order for that to happen, the child has to have both genes, has to have two sickle cell genes.

00:28 So that would be 0 .5 times 0 .5.

00:32 So the probability is 0 .25 that the parents will have a child that has sickle cell.

00:42 Now, part b asks if they have five children, and we know that the probability that they have a child with sickle cell is 0 .25.

00:52 And let's say x is the number of children that have sickle cell of those 5.

01:00 We want to know on this one, what's the probability that at least one of them has sickle cell.

01:07 And we know that that is complementary to none of the children having sickle cell.

01:13 And so that means this is a binomial setting, and we always have that combination and a vibe choose, and we don't want any of them to have sickle cell.

01:21 Well, sickle cell is 0 .25, and we don't want any of those.

01:25 That's a 0 .75.

01:26 And we have 0 .75, and these would be children that wouldn't have.

01:30 The trait and that would be to the fifth power and when we do that calculation we end up getting 0 .76 to 7 so i have about a 76 % chance of having at least one child having sickle cell.

01:45 Part c we want to get the probability distribution for this setting and so we know that this is going to be that combination of 5 choose i times 0 .25 to the i and 0 .75 to the 5 to the 5 .5 minus i.

02:04 And we can use our binomial pdf for this.

02:08 And let's quick just set up the table.

02:11 And we know that they could have none, one, two, three, four, and sadly, all five could have a single cell...

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