A card is drawn at random from a standard deck of cards. What is the probability that the card is black or a face card

, which is approximately .

Step-by-step explanation:

A standard playing card deck includes:

The event in question includes a large number of outcomes; it is a compound event. This event includes a large number of outcomes. One way to keep calculations simple is to split this event into two smaller events that are easier to handle: event and event . The choice of event and event should ensure that .

Note that and should be mutually exclusive (i.e., ) to ensure that:

.

One option involves

In other words:

Verify that:

Note that event is itself a compound event with possible outcomes, one for each card in the two black suits. Overall, the event space includes outcomes (one for each card.) Since these outcomes are equally likely:

.

Event is also a compound event. There are two red suits in a standard deck. Each suit includes three face cards. That corresponds to face cards that are not from a black suit. In other words, event

.

Since event and are mutually-exclusive:

.

Therefore:

.

answer: the distance

step-by-step explanation:

to calculate the distance d of a line segment with endpoints (x1, y1) and (x2, y2) use the formula d (x2 x1)2 (y2 y1)2. to calculate the midpoint of a line segment with endpoints (x1, y1) and (x2, y2) use the formula