Brittany is going to ascend a six-step staircase. At any time, she is just as likely to stride up one step or three steps. Find the probability of the event that she will ascend the six steps in the following number of strides. \( \begin{array}{ll}\text { a. two strides } & \text { b. four strides } \\ \text { c. six strides }\end{array} \) \( \begin{array}{l}\text { a. The probability that Brittany will ascend the staircase in two strides is } \frac{1}{4}\end{array} \) (Type an integer or a simplified fraction.) \( \begin{array}{l}\text { b. The probability that Brittany will ascend the staircase in four strides is } \\ \text { (Type an integer or a simplified fraction.) }\end{array} \)

Brittany’s strides can be interpreted as a sequence where in each stride she takes either 1 or 3 steps, with each possibility equally likely. For any sequence of n strides, the total number of outcomes is 2ⁿ (since each stride has 2 equally likely choices). Below are the details for each case: ––––––––––––––––––––––––––––––– Case (a): 2 Strides We need sequences of two strides that add up to 6 steps. Let the strides be s₁ and s₂ (each being 1 or 3). The only possibility is:   3 + 3 = 6 There is only 1 valid sequence and the total number of outcomes is 2² = 4. Thus, the probability is 1/4. (An answer of 1/4 is already given for part (a).) ––––––––––––––––––––––––––––––– Case (b): 4 Strides Let x be the number of 1‐step strides and y be the number of 3‐step strides. Since there are 4 strides:   x + y = 4 The total steps equation is:   1·x + 3·y = 6 Substitute x = 4 – y into the steps equation:   (4 – y) + 3y = 6   4 + 2y = 6   2y = 2  ⟹  y = 1 So, x = 4 – 1 = 3. In other words, Brittany must take exactly one 3‐step stride and three 1‐step strides. The number of valid sequences is the number of ways to choose which one of the 4 strides is the 3‐step, i.e., C(4, 1) = 4. The total number of outcomes for 4 strides is 2⁴ = 16. Therefore, the probability is:   Probability = 4/16 = 1/4 ––––––––––––––––––––––––––––––– Thus, the probability that Brittany will ascend the staircase in four strides is 1/4. (For completeness, if you are also interested in the case of six strides: Case (c): 6 Strides x + y = 6 and 1·x + 3·y = 6   Substitute x = 6 – y:   (6 – y) + 3y = 6 → 6 + 2y = 6 → 2y = 0 → y = 0, hence x = 6. There is only 1 way (all strides are 1-step). The total number of outcomes is 2⁶ = 64. Therefore, the probability is 1/64.) ––––––––––––––––––––––––––––––– Final Answers:  (a) 1/4  (b) 1/4  (c) 1/64 For part (b) specifically, the answer is 1/4.