What are the possibilities? (2023)

key concepts
mathematics
probability
fractions
percent

introduction
Have you ever heard someone say the odds of something happening are "50-50"? What does that really mean? This sentence has something to do with probability. Probability indicates how likely it is that an event will occur. This means that you can calculate the probability of occurrence for certain events. In this activity, you make these calculations and then test them to see if they are correct.

Fonds
Probability allows us to quantify the likelihood of an event occurring. You may be familiar with the words we use to talk about probability such as "certain", "probable", "unlikely", "impossible", etc. You also probably know that the probability of an event occurring is ranging from impossible, meaning that the event will not occur under any circumstances, to certainty, meaning that there is no doubt that an event will occur. In mathematics, these extreme probabilities are expressed as 0 (impossible) and 1 (sure). This means that a probability number is always a number from 0 to 1. The probability can also be written as a percentage, which is a number from 0 to 100 percent. The higher the probability or percentage of an event, the more likely it is that the event will occur.

The probability of a given event occurring depends on how many possible outcomes the event has. When an event has only one possible outcome, the probability of that outcome is always 1 (or 100 percent). However, when there is more than one possible outcome, this changes. A simple example is tossing a coin. When you toss a coin, there are two possible outcomes (heads or tails). As long as the coin hasn't been tampered with, the theoretical probabilities of both outcomes are the same, they are equally probable. The sum of all possible outcomes is always 1 (or 100 percent) because one of the possible outcomes is certain to occur. This means that on a coin toss, the theoretical probability of heads or tails is 0.5 (or 50 percent).

It gets more complicated with a six-sided die. In this case, when you roll the dice, there are 6 possible outcomes (1, 2, 3, 4, 5, or 6). Can you find out the theoretical probability of each number? It's 1/6 or 0.17 (or 17 percent). In this activity, you will test your probability calculations. The interesting thing about probabilities is that knowing the theoretical probability of a given outcome doesn't necessarily tell you anything about the experimental probabilities if you actually try (unless the probability is 0 or 1). For example, outcomes with very low theoretical probabilities will occur in reality, even though they are highly improbable. So how do your theoretical probabilities match your experimental results? You'll find out by tossing and rolling a coin in this activity.

materials

• Currency
• six-sided dice
• Papier
• pen or pencil

preparation

• Prepare a counting sheet to count the number of times the coin has landed heads or tails.
• Prepare a second counting sheet to count the number of times you rolled each number that was rolled.

procedure

• Determine the theoretical probability that a coin will land heads or tails. Write the probabilities as a fraction.What is the theoretical probability for each side?
• Now get ready to flip your coin.How often do you expect heads or tails from the 10 throws?
• Flip the coin 10 times. After each roll, write down on your score sheet whether you got heads or tails.
• Count how many times you got heads and how many times you got tails. Write your results as a fraction. For example, 3 crosses out of 10 rolls would be 3/10 or 0.3. (The denominator is always the number of coin tosses, and the numerator is the outcome you measure, such as how many times the tails coin lands.) You can also express the same results by looking at the falling heads. for the same 10 pitches. That would be 7 heads in 10 throws: 7/10 or 0.7.Do your results meet your expectations?
• Perform 10 more coin tosses.Do you expect the same results? Why or why not?
• Compare your results from the second round with those from the first round.They are equal? Why or why not?
• Keep tossing the coin. Throw it 30 times in a row this time. Record your results for each throw on your tally sheet.What results do you expect this time?
• View your results from the 30 coin tosses and convert them to fraction form.How are they different from your previous results on the 10 coin toss?
• Count how many heads and tails you've gotten so far in the total coin toss, which should be 50. Again, write your results as a fraction (with the number of tosses as the denominator (50) and the result you are counting as the numerator). ). ). ).Does your experimental probability agree with your theoretical probability for the first step?(An easy way to convert this fraction to a percent is to multiply the denominator and numerator by 2, so 50 x 2 = 100. And after multiplying your numerator by 2, you get a number outside of 100: ya percentage).
• Find the theoretical probability of rolling each number on a six-sided die. Write the probabilities as a fraction.What is the theoretical probability of each number?
• Take the dice and roll them 10 times. After each throw, write down what number you got on your counting sheet.How often do you expect each number from the 10 reels?
• After 10 tosses, compare your results (written as a fraction) to your predictions.how close are they
• Roll the dice another 10 times and record the result of each roll.Are your results changing?
• Now roll the dice 30 times in a row (note the result after each roll).How many times did you roll each number this time?
• Count how many times you rolled each number in the combined 50 rolls. Write your results as a fraction.Does your experimental probability agree with your theoretical probability?(Use the same formula you used for the coin toss and multiply the denominator and numerator each by 2 to get the percentage.)
• Compare your calculated probability numbers to your actual data for both activities (coin and dice).What do your combined results tell you about the probability?
• Extra:Keep increasing the number of coin tosses and dice tosses.How do your results compare to the calculated probabilities as the number of events (toss or toss) increases?
• Extra:Look up how probabilities can be represented by probability trees.Can you draw a probability tree for the coin toss and the dice toss?
• Extra:If you're interested in more advanced probability calculations, find out how to calculate the probability of a recurring event, for example:How likely are you to get two heads in a row when tossing a coin?

Observations and Results
Calculating the probabilities of tossing a coin is fairly straightforward. A coin toss has only two possible outcomes: heads or tails. Both outcomes are equally likely. This means that the theoretical probability of getting heads or tails is 0.5 (or 50 percent). The probabilities of all possible outcomes should add up to 1 (or 100 percent), which they do. However, if you tossed the coin 10 times, you most likely didn't get five heads and five tails. In reality, your results could have been 4 heads and 6 tails (or some other result than 5 and 5). These numbers would be your experimental probabilities. In this example it is 4 out of 10 (0.4) for heads and 6 out of 10 (0.6) for tails. If you repeated all 10 coin tosses, you probably got a different result on the second round. The same was probably true for every 30 coin tosses. Even if you add up all 50 coin tosses, chances are you won't end up with a perfectly equal probability of heads and tails. Therefore, your experimental probabilities probably didn't agree with your calculated (theoretical) probabilities.

You have probably already observed a similar phenomenon when throwing the dice. Although the theoretical probability of each number is 1 in 6 (1/6 or 0.17), in reality your experimental probabilities were probably different. Instead of doing each number 17 percent of his total recordings, he may have done them more or less often.

If you kept tossing the coin or rolling the dice, you probably noticed that the more trials (tosses a coin or dice) you made, the closer the experimental probability was to the theoretical probability. In general, these results mean that even if you know the theoretical probabilities of each possible outcome, you can never know what the actual experimental probabilities will be if there is more than one outcome for an event. Finally, a theoretical probability simply predicts the likelihood that a particular event or outcome will occur; it will not tell you what will actually happen!

more to explore
probability, from Math is fun
probability tree diagrams, from Math is fun
Frequency of results in a small number of studies, von Science Buddies
Pick any card, von Science Buddies
STEM activities for kids, von Science Buddies

This activity presented in collaboration withfriends of science