Question

What is the probability that a person who is older than 35 years has a hemoglobin level between 9 and 11? A. 0. 257 B. 0. 284 C. 0. 312 D. 0. 356 E. 0. 548.

Answer 1

The probability that a person who is older than 35 years has a hemoglobin level between 9 and 11 is given by: Option B: 0.284 approx.

How to form two-way frequency table?

Suppose two dimensions are there, viz X and Y. Some values of X are there as $X_1, X_2, ... , X_n$ and some values of Y are there as $Y_1, Y_2, ... , Y_n$

List them in title of the rows and left to the columns. There will be $n \times k$ table of values will be formed(excluding titles and totals), such that:

Value(ith row, jth column) = Frequency for intersection of  $X_i$  and $Y_j$  (assuming X values are going in rows, and Y values are listed in columns).

Then totals for rows, columns, and whole table are written on bottom and right margin of the final table.

For n = 2, and k = 2, the table would look like:

$\begin{array}{cccc}&Y_1&Y_2&\rm Total\\X_1&n(X_1 \cap Y_1)&n(X_1\cap Y_2)&n(X_1)\\X_2&n(X_2 \cap Y_1)&n(X_2 \cap Y_2)&n(X_2)\\\rm Total & n(Y_1) & n(Y_2) & S \end{array}$

where S denotes total of totals, also called total frequency.

n is showing the frequency of the bracketed quantity, and intersection sign in between is showing occurrence of both the categories together.

How to calculate the probability of an event?

Suppose that there are finite elementary events in the sample space of the considered experiment, and all are equally likely.

Then, suppose we want to find the probability of an event E.

Then, its probability is given as

$P(E) = \dfrac{\text{Number of favorable cases}}{\text{Number of total cases}} = \dfrac{n(E)}{n(S)}$

where favorable cases are those elementary events who belong to E, and total cases are the size of the sample space.

The missing frequency table is attached below. If we name events as:

• A = event that the person selected randomly would be older than 35 years
• B = event that the randomly selected person would be having hemoglobin between 9 and 11

Then, the needed probability is P(B|A) (as it is already specified that the person would be older than 35 years.)

Using the table, we get:

$P(A) = \dfrac{\text{Total person who are older than 35 years age}}{\text{Total person in survey}}\\\\P(A) = \dfrac{162}{429} \approx 0.3776$

Using the table, we get the value on the place where A and B both occur(when person selected is older than 35 years and have hemoglobin between 9 and 11 ) as:

$n(A \cap B) = 162 - 40 - 76 = 46$

Thus, we get:

$\\\\P(A\cap B) = \dfrac{\text{Total person older than 35 years age and have hemoglobin level between 9 and 11}}{\text{Total person in survey}}\\\\P(A \cap B) = \dfrac{46}{429} \approx 0.1072$

Using the chain rule, we get:

$P(B|A)P(A) = P(A\cap B)\\\\P(B|A) = \dfrac{P(A \cap B)}{P(A)} = \dfrac{0.1072}{0.3776} \approx 0.284$

Thus, the probability that a person who is older than 35 years has a hemoglobin level between 9 and 11 is given by: Option B: 0.284 approx.

Learn more about probability here:

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