In the previous section, you investigated the differences between independent events and dependent events.

Click below to learn more about independent and dependent events.

Suppose a bag contains several color tiles: 4 red tiles, 3 green tiles, 6 yellow tiles, and 2 blue tiles.

Cynthia reaches into the bag without looking and randomly pulls out one color tile. Use the bag containing the tiles to determine the probability of each color being drawn.

Drag the fraction representing this probability to the appropriate place.

The probabilities that you just identified are for the simple event of one color tile being drawn. Suppose, however, that Cynthia draws two tiles at the same time. Because two simple events will occur, these events become compound events.

Pause and Reflect

When Cynthia draws the first tile, will that affect the possibilities of the tile that she can draw for the second tile? Why or why not?

Interactive popup. Assistance may be required.

Check Your Answer

Yes. When Cynthia draws the first tile, there are only 14 tiles remaining for the second tile.

Are these events independent events or dependent events? How do you know?

Interactive popup. Assistance may be required.

Check Your Answer

These events are dependent events, because the outcome of the second event (drawing the second tile) depends on what happened for the first event (drawing the first tile).

To determine the probability of a set of compound events, you must first identify the probabilities of each of the events occurring by themselves.

Remember the definition of probability.

Probability = Number of Desired Outcomes over Total Number of Possible Outcomes Number of Desired Outcomes Total Number of Possible Outcomes

Let’s go back to the bag of tiles. Cynthia will draw two tiles from the bag at the same time.

Drag the fraction that represents the probability of each of the following events to the space indicated. For example, for the first line, drag the fraction representing the probability of Cynthia drawing a red tile first to the box in the Probability of First Event column. Drag the fraction representing the probability of Cynthia drawing a green tile second to the box in the Probability of Second Event column. You may use some fractions more than once, or not at all.

Now, calculate the probability of both events occurring for each player by multiplying the probability of the first event and second event together.

Check Your Answer

The probability of Cynthia drawing a red tile and a green tile is equal to:
P(red) × P(green) = 4 over 15 4 15 × 3 over 14 3 14 = 12 over 210 12 210 = 2 over 35 2 35

The probability of Cynthia drawing a red tile and a yellow tile is equal to:
P(red) × P(yellow) = 4 over 15 4 15 × 6 over 14 6 14 = 24 over 210 24 210 = 4 over 35 4 35

The probability of Cynthia drawing a yellow tile and a blue tile is equal to:
P(yellow) × P(blue) = 6 over 15 6 15 × 2 over 14 2 14 = 12 over 210 12 210 = 2 over 35 2 35

The probability of Cynthia drawing a green tile and a blue tile is equal to:
P(green) × P(blue) = 3 over 15 3 15 × 2 over 14 2 14 = 6 over 210 6 210 = 1 over 35 1 35

Practice

A word game consists of tiles where each tile has one letter. There are 42 vowels and 26 consonants. The 42 vowel tiles are distributed as shown in the table.

Samuel draws 2 tiles from the bag containing all of the letter tiles.

1. For the first draw, how many total tiles are in the bag? How many of those tiles are vowels?

   Interactive popup. Assistance may be required.

   Need a hint?

   To determine the total number of tiles in the bag, add the number of tiles with vowels to the number of tiles with consonants.

   Interactive popup. Assistance may be required.

   Check Your Answer

   There are 68 tiles with 42 vowels.
2. For the second draw, how many total tiles are in the bag? How many of those tiles are vowels?

   Interactive popup. Assistance may be required.

   Need a hint?

   Assume that Samuel drew a vowel on the first draw. Since he did not replace the first tile, the number of vowels and the number of total tiles have both been reduced by 1.

   Interactive popup. Assistance may be required.

   Check Your Answer

   There are 67 tiles with 41 vowels.
3. What is the probability that he will draw 2 vowels?

   Interactive popup. Assistance may be required.

   Need a hint?

   Multiply the probability of each tile being a vowel.
   Interactive popup. Assistance may be required.

   Check Your Answer

   42 vowels over 68 tiles 42 vowels 68 tiles × 41 vowels over 67 tiles 41 vowels 67 tiles = 1722 over 4556 1722 4556 = 861 over 2278 861 2278 ≈ 37.8%

Ingrid draws 2 tiles from the bag containing all of the letter tiles.

4. What is the probability that Ingrid selected 2 tiles with the letter A?

   Interactive popup. Assistance may be required.

   Need a hint?

   Assume that Ingrid drew a letter A on the first draw. Since she did not replace the first tile, the number of tiles with the letter A and the number of total tiles have both been reduced by 1.
   Interactive popup. Assistance may be required.

   Check Your Answer

   9 A'2 over 68 tiles 9 A's 68 tiles × 8 A's over 67 tiles 8 A's 67 tiles = 72 over 4556 72 4556 = 18 over 1139 18 1139 ≈ 1.6%


A box contains 4 berry, 3 cinnamon, 4 apple, and 5 carob granola bars.

5. Harold randomly selected 2 granola bars from the box without looking. What is the probability that Harold selected 1 berry granola bar and 1 carob granola bar?

   Interactive popup. Assistance may be required.

   Need a hint?

   After the granola bar was selected, how many carob granola bars are remaining to choose from? How many total granola bars are left?

   Interactive popup. Assistance may be required.

   Check Your Answer

   4 over 16 4 16 × 5 over 15 5 15 = 20 over 240 20 240 = 1 over 12 1 12