Jacob uses the following procedure to write down a sequence of numbers. First he chooses the first term to be 6. To generate each succeeding term, he flips a fair coin. If it comes up heads, he doubles the previous term and subtracts 1. If it comes up tails, he takes half of the previous term and subtracts 1. What is the probability that the fourth term in Jacob's sequence is an integer?
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Toolkit + CCSS Solution
Understand
Restated: Jacob builds a sequence. The first term is 6. To get each next term he flips a fair coin: heads means double the current term and subtract 1; tails means halve the current term and subtract 1. Find the chance that the fourth term is a whole number.
Givens: The first term is 6.; Heads rule: the next term is 2 times the current term, minus 1.; Tails rule: the next term is half the current term, minus 1.; The coin is fair, so heads and tails each have probability 1/2.
Unknowns: The probability that the fourth term of the sequence is an integer.
Understand
Restated: Jacob builds a sequence. The first term is 6. To get each next term he flips a fair coin: heads means double the current term and subtract 1; tails means halve the current term and subtract 1. Find the chance that the fourth term is a whole number.
Givens: The first term is 6.; Heads rule: the next term is 2 times the current term, minus 1.; Tails rule: the next term is half the current term, minus 1.; The coin is fair, so heads and tails each have probability 1/2.
Plan
Primary tool: #1 Draw a Diagram
Secondary: #2 Make a Systematic List, #4 Introduce a Variable, #7 Identify Subproblems
Only 3 flips ever happen, so there are just 8 possible paths. A branching tree diagram lays out every path and the fourth term it produces, so we can simply count how many of those land on a whole number.
Execute — Answer: D
#2 Make a Systematic List 7.SP.C.7Step 1
Count the paths
The fourth term is three steps after the first.
Each step is one coin flip with two outcomes, so the number of flip sequences is 2 times 2 times 2 = 8.
Because the coin is fair and the flips are independent, all 8 paths are equally likely.
$$2^3 = 8$$
💡 Three independent heads-or-tails choices always split into 8 equally likely paths.
#4 Introduce a Variable 6.EE.A.2Step 2
Write the two rules
Let x be the current term.
Heads turns x into 2x - 1, and tails turns x into x/2 - 1.
Writing the rules this way lets us apply them quickly to any value, whole or fractional.
$$H:\ x \to 2x - 1, \quad T:\ x \to \frac{x}{2} - 1$$
💡 Naming the current value x turns each coin rule into one reusable formula.
#1 Draw a Diagram 7.NS.A.3Step 3
Grow the tree
Start at 6 and apply both rules at each level.
Term 2: heads gives 11, tails gives 2.
Term 3: from 11 we get 21 or 4.5, and from 2 we get 3 or 0.
Term 4: from 21 we get 41 or 9.5; from 4.5 we get 8 or 1.25; from 3 we get 5 or 0.5; from 0 we get -1 or -1.
[5]
#1 7.SP.C.7The probability equals the number of integer paths divided by the total number o
Review
Reasonableness: 5/8 sits between 1/2 and 3/4, which fits the picture: a bit more than half the paths work. A shortcut confirms the count. Every term-3 value here (21, 4.5, 3, 0) is a multiple of 1/2, so a final heads (2x - 1) always lands on a whole number, covering 4 paths. The only tails ending that is an integer comes from the path tails-tails, which reaches 0, since 0/2 - 1 = -1. That adds 1 more path, giving 5 in all, matching the tree count.
Alternative: Use the complement. A fourth term fails to be an integer only when a halving step creates a leftover fraction. Scanning the tree, the fraction-producing paths are exactly HHT (9.5), HTT (1.25), and THT (0.5), which is 3 paths. Subtracting from the 8 total leaves 8 - 3 = 5 integer paths, so the probability is again 5/8.
CCSS standards used (min grade 7)
7.SP.C.7 Develop probability models and use them to find probabilities of events (Treating the 8 flip paths as equally likely and forming probability as favorable paths over total paths)
6.EE.A.2 Write, read, and evaluate expressions in which letters stand for numbers (Writing the heads and tails coin rules as the expressions 2x - 1 and x/2 - 1)
7.NS.A.3 Solve real-world problems involving the four operations with rational numbers (Computing each term across the tree, including fractional values like 4.5 and 1.25)
7.SP.C.8 Find probabilities of compound events using organized lists, tables, and simulation (Using the tree diagram to enumerate the compound flip outcomes and count integer endings)
⭐ When only a few coin flips can happen, draw the whole tree and just count the branches you want.
⭐ When only a few coin flips can happen, draw the whole tree and just count the branches you want.