An Example of {AND, OR, Given that}

Using a Normal Distribution

By Henry Mesa

Consider the following problem. The length of human pregnancy in days, has an average of 266 days, and a standard deviation of 16 days. The distribution is normal. Let the random variable X denote the length of a human pregnancy.

Consider the following problem. The length of human pregnancy in days, has an average of 266 days, and a standard deviation of 16 days. The distribution is normal. Let the random variable X denote the length of a human pregnancy.

Let event A be a pregnancy lasts between 266 days and 298 days.

Let event B be a pregnancy lasts between 250 days 282 days.

Let event C be a pregnancy lasts 234 days or less.

Consider the following problem. The length of human pregnancy in days, has an average of 266 days, and a standard deviation of 16 days. The distribution is normal. Let the random variable X denote the length of a human pregnancy.

Let event A be a pregnancy lasts between 266 days and 298 days.

Let event B be a pregnancy lasts between 250 days 282 days.

Let event C be a pregnancy lasts 234 days or less.

1. Are events A and B disjoint?

No, they share the common days 266 to 282.

Consider the following problem. The length of human pregnancy in days, has an average of 266 days, and a standard deviation of 16 days. The distribution is normal. Let the random variable X denote the length of a human pregnancy.

Let event A be a pregnancy lasts between 266 days and 298 days.

Let event B be a pregnancy lasts between 250 days 282 days.

Let event C be a pregnancy lasts 234 days or less.

2. Are events B and C disjoint?

Yes, they do not share any common days.

Consider the following problem. The length of human pregnancy in days, has an average of 266 days, and a standard deviation of 16 days. The distribution is normal. Let the random variable X denote the length of a human pregnancy.

Let event A be a pregnancy lasts between 266 days and 298 days.

Let event B be a pregnancy lasts between 250 days 282 days.

Let event C be a pregnancy lasts 234 days or less.

3. Calculate P(B OR C)

P(X < 234 OR 250 < X < 282) =234 266 250 266 282 266

P Z + P Z 16 16 16

= P(Z < -2) + P( -1< Z< 0 )

= 0.025 + 0.34

= 0.365 Using 68-95-99.7 rule

Consider the following problem. The length of human pregnancy in days, has an average of 266 days, and a standard deviation of 16 days. The distribution is normal. Let the random variable X denote the length of a human pregnancy.

Let event A be a pregnancy lasts between 266 days and 298 days.

Let event B be a pregnancy lasts between 250 days 282 days.

Let event C be a pregnancy lasts 234 days or less.

3. Calculate P(B OR C)

P(X < 234 OR 250 < X < 282) =234 266 250 266 282 266

P Z + P Z 16 16 16

= P(Z < -2) + P( -1< Z< 0 )

= 0.02275 + 0.34135

= 0.36410

Consider the following problem. The length of human pregnancy in days, has an average of 266 days, and a standard deviation of 16 days. The distribution is normal. Let the random variable X denote the length of a human pregnancy.

Let event A be a pregnancy lasts between 266 days and 298 days.

Let event B be a pregnancy lasts between 250 days 282 days.

Let event C be a pregnancy lasts 234 days or less.

4. Calculate P(A OR B)

P(250 < X < 282 OR 266 < X < 298) =

250 266 298 266 = P Z

16 16

= P( -1 < Z < 2 )

= 0.815Using 68 –95-99.7 rule

P(250 < X < 298)

= 0.34 + 0.475

Consider the following problem. The length of human pregnancy in days, has an average of 266 days, and a standard deviation of 16 days. The distribution is normal. Let the random variable X denote the length of a human pregnancy.

Let event A be a pregnancy lasts between 266 days and 298 days.

Let event B be a pregnancy lasts between 250 days 282 days.

Let event C be a pregnancy lasts 234 days or less.

4. Calculate P(A OR B)

P(P(250 < X < 282 OR 266 < X < 298) =

250 266 298 266 = P Z

16 16

= P( -1 < Z < 2 )

= 0.3414 +.4773

= 0.8187

P(250 < X < 298)

Consider the following problem. The length of human pregnancy in days, has an average of 266 days, and a standard deviation of 16 days. The distribution is normal. Let the random variable X denote the length of a human pregnancy.

Let event A be a pregnancy lasts between 266 days and 298 days.

Let event B be a pregnancy lasts between 250 days 282 days.

Let event C be a pregnancy lasts 234 days or less.

5. Calculate P(A AND B)

P(250 < X < 282 AND 266 < X < 298) =

266 266 282 266 = P Z

16 16

= P(0 < Z < 1)

= 0.34 Using 68-95-99.7 rule.

= 0.34135

P(266 < X < 282)

Consider the following problem. The length of human pregnancy in days, has an average of 266 days, and a standard deviation of 16 days. The distribution is normal. Let the random variable X denote the length of a human pregnancy.

Let event A be a pregnancy lasts between 266 days and 298 days.

Let event B be a pregnancy lasts between 250 days 282 days.

Let event C be a pregnancy lasts 234 days or less.

5. Calculate P(A AND B)

P(250 < X < 282 AND 266 < X < 298) =

266 266 282 266 = P Z

16 16

= P(0 < Z < 1)

= 0.3413

P(266 < X < 282)

Consider the following problem. The length of human pregnancy in days, has an average of 266 days, and a standard deviation of 16 days. The distribution is normal. Let the random variable X denote the length of a human pregnancy.

Let event A be a pregnancy lasts between 266 days and 298 days.

Let event B be a pregnancy lasts between 250 days 282 days.

Let event C be a pregnancy lasts 234 days or less.

6. Calculate P(B AND C)

P(X < 234 AND 250 < X < 282) = 0

There is no chance that one observation can meet both criteria.

Consider the following problem. The length of human pregnancy in days, has an average of 266 days, and a standard deviation of 16 days. The distribution is normal. Let the random variable X denote the length of a human pregnancy.

Let event A be a pregnancy lasts between 266 days and 298 days.

Let event B be a pregnancy lasts between 250 days 282 days.

Let event C be a pregnancy lasts 234 days or less.

7. Calculate P(A | B)

Given that the pregnancy lasted between 250 and 282 days, there is a 50 % chance that this particular pregnancy lasted between 266 and 298 days.

P(A AND B) =

P(B)

P(266 < X < 282) =

P(250<X<282)

The new whole/sample space.

0.34 =

0.68

= 0.5

Consider the following problem. The length of human pregnancy in days, has an average of 266 days, and a standard deviation of 16 days. The distribution is normal. Let the random variable X denote the length of a human pregnancy.

Let event A be a pregnancy lasts between 266 days and 298 days.

Let event B be a pregnancy lasts between 250 days 282 days.

Let event C be a pregnancy lasts 234 days or less.

8. Calculate P(B | A)

Given that the pregnancy lasted between 266 and 298 days, there is a 71.58 % chance that this particular pregnancy lasted between 250 and 282 days.

P(A AND B) =

P(A)

P(266 < X < 282) =

P(266 < X < 298)

The new whole/sample space.

0.34 =

0.475

= 0.7158