Name______

Date______

Patterns in THEORETICAL PROBABILITY

MYP 3 – Math Investigation

Assessment: Criterion B

Rubric for Criterion B – Investigating Patterns

Level of Achievement / Descriptor
0 / The student does not reach a standard described by any of the descriptors given below.
1 – 2 / The student applies, with some guidance, mathematical problem solving techniques to recognize simple patterns.
3 – 4 / The student selects and applies mathematical problem solving techniques to recognize patterns, and suggests relationships or general rules.
5 – 6 / The student selects and applies mathematical problem solving techniques to recognize patterns, describes them as relationships or general rules, and draws conclusions consistent with findings.
7 – 8 / The student selects and applies mathematical problem solving techniques to recognize patterns, describes them as relationships or general rules, and draws conclusions consistent with findings, and provides justifications or proofs.

Assessment Task:

1)  Complete the following tasks and questions looking for any patterns. Show all your work!

2)  Write a general rule or formula in mathematical language to assist in extending a problem.

3)  Justify (Explain or Prove) that your rule works for all examples

4)  Show all probabilities as UNSIMPLIFIED FRACTIONS

PART I: COLLECTING DATA

TOSSING COINS

Create a tree diagram to show the sample space of tossing a coin 4 times then answer the questions below:

(i) You toss a coin one time

1. List the sample space ______

2. The number of events in the sample space is ______

3. The probability of getting 1 head is ______

(ii) You toss a coin two times

4. List the sample space ______

5. The number of events in the sample space is ______

6. The probability of getting 2 heads is ______

(iii) You toss a coin three times

7. List the sample space ______

8. The number of events in the sample space is ______

9. The probability of getting 3 heads is ______

(iv) You toss a coin four times

10. List the sample space ______

11. The number of events in the sample space is ______

12. The probability of getting 4 heads is ______

PART II: ORGANIZING DATA

13. Organize your results into a table that shows the relationship between:

·  Number of coins

·  Number of events in the sample space

·  Probability of getting all heads

Number of Coins / Number of Events in
Sample Space / Probability of Getting
All Heads

14. Describe mathematically or in words what you think the relationship is between the number of coins tossed and the probability of getting all heads. Be specific, comment of each situation of tossing a coin 1, 2, 3, or 4 times and how that relates to the probability.

PART III: FINDING A GENERAL RULE FOR COINS

15. Complete the following statements based on the pattern you have observed, without a tree diagram.

a) If a coin was tossed 5 times, the probability of getting 5 heads would be: ______

b) If a coin was tossed 6 times, the probability of getting 6 heads would be: ______

c) If a coin was tossed 7 times, the probability of getting 7 heads would be: ______

d) If a coin was tossed 10 times, the probability of getting 10 heads would be: ______

16. Explain how you used the pattern to answer question 15 a,b,c,d

17. State your GENERAL RULE for finding the probability of getting ‘x’ heads when tossing a coin ‘x’ times:

18. Justify (explain) why your rule works for any number of times a coin is tossed

PART IV: APPLICATION AND GENERALISING OF THE RULE

Use a version of your rule to answer each of the following questions

19.

a) If you are given 3 colored discs placed in a hat: 1 blue, 1 yellow, and 1 red; what is the probability that you select the yellow disc if you draw only 1 disc?

b) What is the probability that you select the yellow disc 4 times in a row?

c) How can what you learned with tossing coins be applied to this situation?

20.

a) What is the probability of a mother giving birth to a girl?

b) What is the probability of a mother giving birth to 5 girls in a row?

c) How can what you learned with tossing coins be applied to this situation?

21.

a) What is the probability of rolling a 6 with one toss of a die?

b) What is the probability of rolling a 6 three times in a row?

c) How can what you learned with tossing coins be applied to this situation?

21. In General, if the probability of an event occurring once is the probability of that event happening ‘b’ times in a row is:

PART V: EXTENSION (Optional)

1. Tossing a coin is called a binomial ‘event’ because there are only 2 possible outcomes, H or T.

a) Write down the sample space for tossing two coins:

i) List, in order, the probability of tossing 2 heads, 1 head, 0 heads

b) Write down the sample space for tossing three coins:

ii) List, in order, the probability of tossing 3 heads, 2 heads, 1 head, 0 heads

c) Write down the sample space for tossing four coins:

ii) List, in order, the probability of tossing 4 heads, 3 heads, 2 heads, 1 head, 0 heads

2. How can you quickly figure out the denominator in each case?

3. How does the sum of the numerators in each example relate to the denominators? Why?

4. Look closely at the numerators, what familiar pattern do you see? How could this help you?

5. Using this, what is the probability of getting 4 heads when tossing a coin 6 times? Explain how you arrived at this result and demonstrate your process.

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