Question

The Masterfoods company says that before the introduction of purple, yellow candies made up \(20 \%\) of their plain M\&M's, red another \(20 \%\), and orange, blue, and green each made up \(10 \%\). The rest were brown. a) If you pick an M\&M at random, what is the probability that 1) it is brown? 2) it is yellow or orange? 3) it is not green? 4) it is striped? b) If you pick three M\&M's in a row, what is the probability that 1) they are all brown? 2) the third one is the first one that's red? 3) none are yellow? 4) at least one is green?

Short answer

a) 1) 0.3; 2) 0.3; 3) 0.9; 4) 0 b) 1) 0.027; 2) 0.128; 3) 0.512; 4) 0.271

Step-by-step solution

Calculate Probability of Individual Colors

First, identify the probability of each individual color using the given percentages. Yellow, red: \(20\%\), Orange, blue, green: \(10\%\) each. The remaining probability will go to brown. This is calculated as: \[P(\text{brown}) = 1 - (0.2 + 0.2 + 0.1 + 0.1 + 0.1) = 0.3\] Thus, the probability of picking a brown is \(30\%\).

Calculate Probability for Composite Events in Part a

1) The probability it is brown is already found as \(30\%\) or \(0.3\).2) The probability it is yellow or orange is: \[P(\text{yellow or orange}) = P(\text{yellow}) + P(\text{orange}) = 0.2 + 0.1 = 0.3\]3) The probability it is not green: \[P(\text{not green}) = 1 - P(\text{green}) = 1 - 0.1 = 0.9\]4) Probability it is striped is zero since striped is not mentioned as an option.

Calculate Probability for First Compound Event in Part b

1) Probability all three are brown:\[P(\text{3 brown}) = P(\text{brown}) \times P(\text{brown}) \times P(\text{brown}) = 0.3 \times 0.3 \times 0.3 = 0.027\]

Calculate Probability for Second Compound Event in Part b

2) Probability the third one is the first red:\[P(\text{first red third}) = (1 - P(\text{red}))^2 \times P(\text{red}) = 0.8 \times 0.8 \times 0.2 = 0.128\]

Calculate Probability for Third Compound Event in Part b

3) Probability no yellow:\[P(\text{none yellow}) = (1 - P(\text{yellow}))^3 = 0.8 \times 0.8 \times 0.8 = 0.512\]

Calculate Probability for Fourth Compound Event in Part b

4) Probability at least one is green:\[P(\text{at least one green}) = 1 - P(\text{none green}) = 1 - (0.9)^3 = 1 - 0.729 = 0.271\]

Key concepts

Compound Events

In probability, compound events involve multiple events happening at the same time or in sequence. Unlike single events, which only concern one outcome, compound events can be more complex. For example, in the Masterfoods M&M exercise, the compound event consists of picking several M&Ms in succession.

• Compound events can be categorized as dependent or independent. This affects how their probabilities are calculated.
• In this exercise, picking multiple M&Ms involves compound events because the sequence of selections matters.

Whether you're calculating the probability of picking several candies in a row or specific outcomes, understanding compound events is crucial for solving complex probability scenarios.

Probability Calculation Steps

Calculating probability can sometimes seem daunting, but breaking it down into steps simplifies the process. Here's how you can easily calculate the probability for different scenarios using the M&M exercise as an example:
1. **Identify individual probabilities:** Start by finding the probability of each single event happening. For example, purple M&Ms are not considered here, so first, calculate probabilities for colors mentioned, such as yellow, which is 20% or 0.2.
2. **Use the addition rule for probabilities of 'either/or' events:** If you want the probability of an M&M being yellow or orange, add their separate probabilities, like in this case: 0.2 + 0.1.
3. **Use multiplication for 'and' events:** For compound events like picking three brown M&Ms in a row, multiply the probability of brown times three (0.3 x 0.3 x 0.3).
Breaking it down with these simple steps makes it easier to solve even complex probability problems.

Complement Rule

The complement rule is a handy tool in probability. It allows for finding the likelihood of an event not happening by using the probability of the event occurring. It holds value especially when calculating complex scenarios or when missing direct probability data.

• Here's the basic formula: \[ P(\text{not A}) = 1 - P(A) \]
• In the exercise, this method helps determine the chances of picking an M&M that is not green. Since the probability of selecting a green M&M is given as 10% (or 0.1), the complement rule helps find the probability of not picking a green one: \[ P(\text{not green}) = 1 - P(\text{green}) = 0.9 \]

Using the complement rule simplifies the calculations, especially when dealing with large sample sizes or multiple event possibilities.

Independent Events

Independent events are those where the outcome of one event does not influence the outcome of another. In probability, this can simplify the calculation process significantly. Understanding independent events can unlock a deeper grasp when dealing with repeated trials or selections.

• An example from the exercise is drawing M&Ms multiple times without replacement, assuming infinite supply, where each draw remains independent.
• When calculating probabilities for independent events, such as successive brown picks: \[ P(A\text{ and }B) = P(A) \times P(B) \text{, if A and B are independent} \]
• For instance, finding the probability that all three M&Ms are brown involves multiplying the individual probability of picking a brown M&M three times (0.3 each draw).

Recognizing whether events are independent is crucial, as it dictates how their probabilities interact, streamlining more complex calculations.