A quick analysis of problems that require students to determine whether a given outcome is certain, probable or impossible makes a simple warmup introduction to the subject of sixth-grade probability. Create pictures of spinners with colored sections. For example, show students a spinner that has five blue sections, two red sections and one yellow section. Ask students, "If you spin the spinner once, is it certain, probable or impossible that it will land on blue?" or "How likely is it that the spinner will land on purple?" Students should be able to quickly ascertain that the first instance is probable and the second impossible since there are no purple spaces. A solid color spinner can demonstrate the concept of certainty. Generate similar scenarios using a jar of colored marbles or a deck of cards.
Tossing a coin, rolling dice, drawing numbers and using colored spinners and marbles are easy ways to review simple probability. Toss, roll, spin or draw 10 times according to the demonstration device you are using. Tally the results showing the frequency of each possible result. Instead of simply indicating a general likelihood of a particular outcome, notate the formal probability of each result as: P(result) equals number of occurrences a single result/total number of possible occurrences. For example, you roll a die 10 times and get three fives, two fours, one six and four threes. According to these results, the probability of obtaining a result of five on any given roll is P(5) equals 3/10. Try the experiment with 100 instances and compare results. Give students a word problem describing a spinner and ask them to draw the spinner and write the the probability notation for landing on each color. Describe the colors and numbers of a jar of marbles and ask how many marbles of which colors should be added or subtracted to create a given probability of a certain outcome.
With simple probability under their belts, sixth-graders can tackle problems that take into account whether the outcome is independent or dependent on other factors. For example, in a coin flip, the result of one flip does not influence the next. The probability of heads or tails is 50/50 every time, so each flip is an independent event. Likewise, if you draw numbers from a bag and place each number back in the bag before drawing again, the probability ratios remain the same on each draw. However, if you remove a selection without replacing it each time, the probability of the results of each successive draw is dependent on the preceding draws because the range of possibilities changes each time. Permutation and combination problems will demonstrate the changing probabilities of dependent events. Give each student a bag of multicolored chocolate candies. Have them count up the total number of candies and each color. Write the simple probability of drawing each color on the first draw. Pour the candies into a brown bag, draw one and recalculate the probabilities based on what is left in the bag. Ask questions such as, "What is the probability of drawing a green, yellow, blue and orange candies in any order in four draws?" or "What is the probability of drawing red, green and yellow, in that order, on the first three draws?"
Sixth-grade students should be able to understand that what is theoretically possible does not always pan out that way under real-world conditions. Theoretical probability tells you that the chances of heads or tails on a coin flip are even odds every time, but in 100 flips, you will not necessarily get 50 heads and 50 tails every time. So the theoretical probability may state P(heads) equals 50/100 but repeated experiments may show P(heads) equals 70/100 or 48/100. The experimental probability will change with every experiment because of the random nature of actual results. Set up an independent probability scenario using dice or colored cubes. Ask students to identify the theoretical probability of each possible outcome, such as P(5) equals 1/6 on a die or P(purple) equals 8/20 in a bag of colored cubes. Roll the die or draw a cube (replacing the cube before the next draw) 100 times and tally the results. Rewrite the probabilities based on the actual results of the experiment. Repeat the experiment several times to compare results and calculate the overall probabilities of the combined experiments to analyze whether the trend over time approximates the theoretical probability.