Lisa owns a ＂Random Candy＂ vending machine, which is a machine that picks a candy out of an assortment in a random fashion. Lisa controls the probability of picking each candy. The machine has too much of the candy ＂Coffee Toffee,＂ so Lisa wants to program it so that the probability of getting ＂Coffee Toffee＂ twice in a row is greater than $\frac{4}{3}$ times the probability of getting a different candy in one try. Write an inequality that models the situation. Use $p$ to represent the probability of getting ＂Coffee Toffee＂ in one try.

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Grade 6

One-step inequalities: word problems

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Q. Lisa owns a ＂Random Candy＂ vending machine, which is a machine that picks a candy out of an assortment in a random fashion. Lisa controls the probability of picking each candy. The machine has too much of the candy ＂Coffee Toffee,＂ so Lisa wants to program it so that the probability of getting ＂Coffee Toffee＂ twice in a row is greater than

\frac{4}{3}

times the probability of getting a different candy in one try. Write an inequality that models the situation. Use

p

to represent the probability of getting ＂Coffee Toffee＂ in one try.

Define Probability of Coffee Toffee: Let $p$ be the probability of getting ＂Coffee Toffee＂ in one try. The probability of getting ＂Coffee Toffee＂ twice in a row is $p^2$ .
Calculate Probability of Different Candy: Let $q$ be the probability of getting a different candy in one try. Since the total probability must sum to $1$ , we have $q = 1 - p$ .
Set Up Inequality: The problem states that the probability of getting ＂Coffee Toffee＂ twice in a row should be greater than $\frac{4}{3}$ times the probability of getting a different candy in one try. So, we set up the inequality: $p^2 > \left(\frac{4}{3}\right) \times q$ .
Substitute and Simplify: Substitute $q = 1 - p$ into the inequality: $p^2 > \frac{4}{3} \cdot (1 - p)$ .
Rearrange Inequality: Simplify the inequality: $p^2 > \frac{4}{3} - \frac{4}{3}p$ .
Rearrange Inequality: Simplify the inequality: $p^2 > \frac{4}{3} - \frac{4}{3}p$ . Rearrange the inequality to bring all terms to one side: $p^2 + \frac{4}{3}p - \frac{4}{3} > 0$ .