1. Cmdr's coffee shop offers a coffee special with coffee syrups of vanilla, chocolate, and caramel. You can also choose between a latte, and cappuccino. How many different coffees can you make for the special?
Ans: 6 (VLatte, ChocLatte, CLatte; VCap, ChocCap, CCap)
2. Edwin rolls a number cube and then spins a color from a color card (red, yellow, blue, green, white). What is the probability that he will roll an even number and choose a red color?
Ans:
cube 3/6 or 1/2, color 1/5
1/2 + 1/5 = 5/10 + 2/10
= 7/10 or 70%??
3. A bag contains 5 blue marbles and 3 green marbles. What is the probability of drawing a blue marble followed by a green marble, with replacing the first marble before drawing the second marble?
Ans: P(blue) = 5/8 = 62.5%??
4. Explain mathematically whether the experiments below have an equal chance or not.
a. Choose a number at random from 1 to 5
work: P(number) = 1/5
b. Toss a coin
work: P(coin) = 1/2
c. Choose a letter at random from the word MOOSE.
work: P(letter) = 1/5
Ans: Experiments a and c have an equal chance because both have a probablity of 1/5 or 20%.
4. Edwin tossed a number cube several times. He got '3' on 4 of the tosses. Based on theoretical probabilities, what is the best estimate of the total number of times he tossed the cube?
Ans: Not sure how to solve.
1. correct
For the rest, think about:
2 & 3 Probability of two independent events:
for two independent events A and B to both happen, the probability is P(A)*P(B)
Example:
The probability of throwing a 4 from a number cube AND drawing a red card from a standard deck is
(1/6)*(1/2)=1/12
4. Equiprobability
If each of the possible outcomes of an experiment has equal probability, the experiment is an equiprobability experiment.
Examples of equiprobability experiments are:
- throwing a number cube (each outcome has probability 1/6)
- tossing a fair coin (each outcome has probability 1/2)
- picking a marble from a bag containing 2 each of colours red, green, blue, black and white.
An example of non-equiprobability experiments:
- picking randomly a digit from the number 10334
P(1)=1/5
P(0)=1/5
P(3)=2/5
P(4)=1/5
so the probabilities are not equal
4. Theoretical probability
Theoretical probability is what we would get if an experiment is repeated many times, if all the assumptions of independence, fairness, etc. are indeed true.
For example, the theoretical probability of getting heads in tossing a coin is 1/2.
If I toss 20 times, the expected number of heads is 20*(1/2)=10.
Conversely, if the expected number of heads is 10, the most probable number of tosses is 10/(1/2)=20, using theoretical probabilities.
Please feel free to post your corrections for checking if you wish.
I'm posting corrections. Still not sure how to answer last question.
2. Edwin rolls a number cube and then spins a color from a color card (red, yellow, blue, green, white). What is the probability that he will roll an even number and choose a red color?
Ans: P(even)*P(red)
= (1/2)*(1/5)
= 1/10 = 10%
3. A bag contains 5 blue marbles and 3 green marbles. What is the probability of drawing a blue marble followed by a green marble, with replacing the first marble before drawing the second marble?
Ans: P(blue)*P(green)
= (5/8)*(3/8)
= 15/64
= 23%
4. Edwin tossed a number cube several times. He got number '3' on 4 of the tosses. Based on theoretical probabilities, what is the best estimate of the total number of times he tossed the cube?
Ans: 3/(1/6) = 18 tosses?
To answer the first question, you can use the concept of combinations to find the number of different coffees that can be made. Since there are three syrup options (vanilla, chocolate, and caramel) and two coffee options (latte and cappuccino), you can multiply the number of options for each category. So, there are 3 choices for syrups and 2 choices for coffee types, resulting in 3 * 2 = 6 different coffees that can be made for the special.
For the second question, you need to find the probability of rolling an even number and choosing a red color separately, and then multiply those probabilities to find the probability of both events happening together. Since there are 6 faces on a number cube and 3 even numbers (2, 4, and 6), the probability of rolling an even number is 3/6 or 1/2. Similarly, since there are 5 color options and 1 red color, the probability of choosing a red color is 1/5. To find the probability of both events happening together, you multiply the probabilities: (1/2) * (1/5) = 1/10 or 10%.
In the third question, if the first marble is replaced before drawing the second marble, then the probability of drawing a blue marble remains the same for both draws. So, the probability of drawing a blue marble is 5/8 or 62.5%.
For the fourth question (part a, b, and c), you can determine whether the experiments have an equal chance or not by comparing their respective probabilities. Experiment a involves choosing a number at random from 1 to 5, which means there are 5 equally likely outcomes, resulting in a probability of 1/5 or 20%. Experiment c involves choosing a letter at random from the word MOOSE, which has 5 letters. Again, there are 5 equally likely outcomes, resulting in a probability of 1/5 or 20%. Therefore, experiments a and c have equal chances.
For the last question, the best estimate of the total number of times Edwin tossed the number cube can be found by considering the fact that he got '3' on 4 of the tosses. Since the probability of getting a '3' on any given toss is 1/6, you can set up a proportion: 4/total number of tosses = 1/6. Cross-multiplying, you get 4 * 6 = 1 * total number of tosses, which simplifies to 24 = total number of tosses. Therefore, the best estimate of the total number of times he tossed the cube is 24.
I hope this explanation helps you understand how to solve these probability problems!