Sometimes, when you have a series of compound events, the outcome of the first event does affect the outcomes of the subsequent events. These events are called dependent events since the outcome of the second (or third) event depends on the outcome of the first event.
Suppose a bag contains several color tiles: 6 red tiles, 4 green tiles, 3 yellow tiles, and 2 blue tiles.
Ramona 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 Ramona draws two tiles at the same time. Because two simple events will occur, these events become compound events.
However, unlike the spinner in the previous section of the lesson, the outcomes of these compound events (two color tiles being drawn at the same time) are dependent on each other. That is, the results of the second tile that is drawn depend on the results of what happened when the first tile was drawn. Since that is the case, we can call these events dependent events.
To determine the probability of a set of dependent events, we must first identify the probabilities of each of the events occurring by themselves. However, the circumstances describing the second event depend on what happened the first time.
In the case of drawing objects, once the first tile has been selected, there are fewer possible outcomes for the second tile (one less). So, we must reduce the denominator in the probability of the second tile by one.
Ramona 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 Ramona drawing a red tile first to the box in the Probability of First Event column. Drag the fraction representing the probability of Ramona drawing a green tile second to the box in the Probability of Second Event column. Hint: The number of tiles left after the first tile has been selected has decreased by one. 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.
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The probability of Ramona drawing a red tile and a green tile is equal to (probability of drawing a red tile) × (probability of drawing a green tile) = 6 over 15 6 15 x 4 over 14 4 14 = 24 over 210 24 210 = 4 over 35 4 35
The probability of Ramona drawing a red tile and a yellow tile is equal to (probability of drawing a red tile) × (probability of drawing a yellow tile) = 6 over 15 6 15 x 3 over 14 3 14 = 18 over 210 18 210 = 3 over 35 3 35
The probability of Ramona drawing a yellow tile and a blue tile is equal to (probability of drawing a yellow tile) × (probability of drawing a blue tile) = 3 over 15 3 15 x 2 over 14 2 14 = 6 over 210 6 210 = 1 over 35 1 35
The probability of Ramona drawing a green tile and a blue tile is equal to (probability of drawing a green tile) × (probability of drawing a blue tile) = 4 over 15 4 15 x 2 over 14 2 14 = 8 over 210 8 210 = 4 over 105 4 105
Hot Tip!
You will also see the words “without replacement” in problems involving the drawing of two objects. This phrase is a clue that you have a situation with dependent events.
Ms. Dawson’s 1st period science class has 18 girls and 12 boys. Ms. Dawson needs to select a pair of students to help her set up a laboratory investigation. She has decided to randomly select 2 students from the entire class.
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After the first girl is selected, how many girls are remaining? How many total students are available to be selected from for the second student?
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18 girls over 30 students
18 girls
30 students
x 17 girls over 29 students
17 girls
29 students
= 306 over 870
306
870
= 51 over 145
51
145 ≈ 35.2%
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After the first boy is selected, how many boys are remaining? How many total students are available to be selected from for the second student?
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12 boys over 30 students
12 boys
30 students
x 11 boys over 29 students
11 boys
29 students
= 132 over 870
132
870
= 22 over 145
22
145 ≈ 15.2%
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After the first gumball is selected, how many green gumballs
are remaining?
How many total gumballs are left?
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15 over 50 15 50 x 14 over 49 14 49 = 210 over 2450 210 2450 = 3 over 35 3 35
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After the first gumball is selected, how many total gumballs are left?
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14 over 50 14 50 x 21 over 49 21 49 = 294 over 2450 294 2450 = 3 over 25 3 25