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James used a table to determine the sample space for an experiment in which a purple (P) or yellow (Y) marble is selected from a bag 4 times and replaced after each draw.

Row 1 PPPP PPPY PPPY PPYP
Row 2 PYPP YPPP PYPY YPYP
Row 3 YYYY YYYP YYPY YPYY
Row 4 PYYY YYPP PPYY PYYP
In which row did James make an error when he created his table? Answer 1 for row 1, 2 for row 2, 3 for row 3, or 4 for row 4.

(1 point)
Row

1 answer

To determine if there is an error in James' table, we need to analyze the combinations of marble selections for each row. Since the experiment involves selecting a marble 4 times with replacement and each marble can either be purple (P) or yellow (Y), the total number of possible outcomes is \(2^4 = 16\).

Let's examine each row and count the unique combinations:

Row 1:

  • PPPP
  • PPPY (listed twice)
  • PPYP

Unique outcomes for Row 1: PPPP, PPPY, PPYP (3 unique combinations)

Row 2:

  • PYPP
  • YPPP
  • PYPY
  • YPYP

Unique outcomes for Row 2: PYPP, YPPP, PYPY, YPYP (4 unique combinations)

Row 3:

  • YYYY
  • YYYP
  • YYPY
  • YPYY

Unique outcomes for Row 3: YYYY, YYYP, YYPY, YPYY (4 unique combinations)

Row 4:

  • PYYY
  • YYPP
  • PPYY
  • PYYP

Unique outcomes for Row 4: PYYY, YYPP, PPYY, PYYP (4 unique combinations)

Now, let's summarize:

  • Row 1 has repeated outcomes (PPPY is counted twice). This indicates that there is a mistake in this row.
  • Rows 2, 3, and 4 have unique outcomes without repetition.

Thus, the row in which James made an error is:

Row 1

So, the answer is 1 (Row 1).