Goat or Car? Game Show Simulator

A player is presented with three doors. Behind one door is a new car. The other two doors conceal a goat. The player chooses a door then the host reveals a goat behind one of the two remaining doors. Should the player maintain his choice or switch to the only remaining door?

My intuition suggested that once the other goat was revealed that it became a simple 50% chance that the player had chosen correctly and, therefore, there would be no statistical benefit in changing their mind. In the simulation below, the player always sticks with their original choice.

Player won 3 times and lost 7 times.

1. Car
2. Car
3. Goat
4. Goat
5. Goat
6. Goat
7. Car
8. Goat
9. Goat
10. Goat

Every refresh of this page runs the test again so you will see variation in results. Of course, 10 iterations are statistically insignificant. A simulation of 1,000 iterations makes the ratio clearer:

Player won 332 times and lost 668 times.

1. Goat
2. Goat
3. Goat
4. Car
5. Goat
6. Goat
7. Goat
8. Goat
9. Car
10. Car
11. Car
12. Car
13. Car
14. Car
15. Car
16. Goat
17. Goat
18. Goat
19. Goat
20. Goat
21. Goat
22. Goat
23. Goat
24. Goat
25. Goat
26. Goat
27. Goat
28. Goat
29. Goat
30. Goat
31. Goat
32. Car
33. Car
34. Car
35. Car
36. Car
37. Car
38. Goat
39. Car
40. Goat
41. Goat
42. Goat
43. Goat
44. Goat
45. Goat
46. Car
47. Goat
48. Goat
49. Goat
50. Goat
51. Goat
52. Goat
53. Goat
54. Goat
55. Car
56. Car
57. Goat
58. Goat
59. Goat
60. Goat
61. Car
62. Car
63. Car
64. Goat
65. Goat
66. Goat
67. Car
68. Goat
69. Goat
70. Goat
71. Car
72. Goat
73. Goat
74. Car
75. Car
76. Goat
77. Goat
78. Car
79. Car
80. Goat
81. Goat
82. Car
83. Goat
84. Goat
85. Goat
86. Car
87. Car
88. Goat
89. Goat
90. Goat
91. Goat
92. Car
93. Goat
94. Goat
95. Goat
96. Goat
97. Goat
98. Car
99. Goat
100. Goat
101. Goat
102. Car
103. Goat
104. Goat
105. Goat
106. Goat
107. Goat
108. Goat
109. Goat
110. Goat
111. Goat
112. Goat
113. Goat
114. Goat
115. Goat
116. Goat
117. Goat
118. Goat
119. Goat
120. Car
121. Goat
122. Goat
123. Car
124. Car
125. Goat
126. Goat
127. Goat
128. Car
129. Car
130. Goat
131. Car
132. Car
133. Goat
134. Car
135. Goat
136. Goat
137. Car
138. Goat
139. Car
140. Goat
141. Car
142. Goat
143. Goat
144. Goat
145. Goat
146. Car
147. Goat
148. Goat
149. Car
150. Goat
151. Goat
152. Car
153. Goat
154. Goat
155. Goat
156. Car
157. Car
158. Goat
159. Goat
160. Goat
161. Goat
162. Car
163. Goat
164. Goat
165. Car
166. Goat
167. Car
168. Car
169. Car
170. Goat
171. Goat
172. Goat
173. Goat
174. Goat
175. Goat
176. Goat
177. Goat
178. Car
179. Goat
180. Goat
181. Car
182. Goat
183. Goat
184. Goat
185. Goat
186. Goat
187. Goat
188. Goat
189. Car
190. Goat
191. Goat
192. Goat
193. Goat
194. Goat
195. Goat
196. Goat
197. Car
198. Car
199. Goat
200. Goat
201. Goat
202. Car
203. Car
204. Goat
205. Car
206. Goat
207. Goat
208. Goat
209. Car
210. Car
211. Goat
212. Car
213. Goat
214. Car
215. Goat
216. Goat
217. Goat
218. Goat
219. Car
220. Goat
221. Car
222. Car
223. Car
224. Goat
225. Car
226. Car
227. Car
228. Car
229. Goat
230. Car
231. Goat
232. Car
233. Goat
234. Goat
235. Goat
236. Car
237. Car
238. Goat
239. Goat
240. Goat
241. Goat
242. Goat
243. Goat
244. Goat
245. Goat
246. Car
247. Car
248. Goat
249. Goat
250. Car
251. Goat
252. Car
253. Goat
254. Car
255. Car
256. Goat
257. Goat
258. Goat
259. Car
260. Car
261. Goat
262. Car
263. Goat
264. Goat
265. Goat
266. Car
267. Goat
268. Car
269. Goat
270. Goat
271. Goat
272. Goat
273. Goat
274. Goat
275. Goat
276. Car
277. Goat
278. Car
279. Goat
280. Car
281. Goat
282. Car
283. Goat
284. Goat
285. Car
286. Goat
287. Goat
288. Car
289. Goat
290. Goat
291. Car
292. Goat
293. Goat
294. Goat
295. Goat
296. Goat
297. Goat
298. Car
299. Goat
300. Goat
301. Goat
302. Goat
303. Car
304. Goat
305. Goat
306. Car
307. Car
308. Goat
309. Goat
310. Car
311. Goat
312. Car
313. Car
314. Goat
315. Car
316. Goat
317. Car
318. Goat
319. Goat
320. Car
321. Car
322. Goat
323. Goat
324. Goat
325. Goat
326. Goat
327. Goat
328. Goat
329. Goat
330. Goat
331. Goat
332. Goat
333. Car
334. Goat
335. Goat
336. Car
337. Goat
338. Goat
339. Goat
340. Car
341. Goat
342. Car
343. Car
344. Goat
345. Goat
346. Goat
347. Goat
348. Car
349. Goat
350. Goat
351. Car
352. Goat
353. Goat
354. Goat
355. Goat
356. Goat
357. Goat
358. Goat
359. Goat
360. Goat
361. Goat
362. Goat
363. Goat
364. Goat
365. Goat
366. Goat
367. Car
368. Car
369. Goat
370. Car
371. Goat
372. Goat
373. Goat
374. Goat
375. Goat
376. Goat
377. Goat
378. Car
379. Car
380. Goat
381. Car
382. Car
383. Goat
384. Goat
385. Goat
386. Goat
387. Goat
388. Goat
389. Goat
390. Goat
391. Goat
392. Goat
393. Goat
394. Goat
395. Goat
396. Goat
397. Goat
398. Goat
399. Goat
400. Goat
401. Car
402. Car
403. Car
404. Goat
405. Car
406. Goat
407. Car
408. Goat
409. Goat
410. Goat
411. Goat
412. Car
413. Goat
414. Car
415. Goat
416. Car
417. Goat
418. Car
419. Car
420. Car
421. Goat
422. Car
423. Car
424. Goat
425. Goat
426. Car
427. Goat
428. Car
429. Goat
430. Goat
431. Car
432. Goat
433. Goat
434. Car
435. Car
436. Goat
437. Goat
438. Goat
439. Goat
440. Goat
441. Goat
442. Car
443. Goat
444. Goat
445. Goat
446. Car
447. Goat
448. Goat
449. Goat
450. Goat
451. Car
452. Goat
453. Goat
454. Goat
455. Goat
456. Goat
457. Goat
458. Goat
459. Car
460. Goat
461. Goat
462. Car
463. Goat
464. Car
465. Goat
466. Goat
467. Car
468. Goat
469. Car
470. Goat
471. Goat
472. Car
473. Car
474. Goat
475. Car
476. Goat
477. Car
478. Goat
479. Goat
480. Goat
481. Goat
482. Car
483. Car
484. Goat
485. Goat
486. Car
487. Goat
488. Car
489. Goat
490. Goat
491. Car
492. Goat
493. Car
494. Car
495. Car
496. Goat
497. Car
498. Goat
499. Goat
500. Car
501. Goat
502. Goat
503. Goat
504. Goat
505. Goat
506. Goat
507. Goat
508. Car
509. Goat
510. Goat
511. Goat
512. Goat
513. Car
514. Goat
515. Car
516. Goat
517. Goat
518. Car
519. Car
520. Car
521. Goat
522. Goat
523. Goat
524. Car
525. Goat
526. Car
527. Car
528. Goat
529. Car
530. Car
531. Goat
532. Goat
533. Car
534. Goat
535. Car
536. Car
537. Goat
538. Car
539. Goat
540. Goat
541. Car
542. Goat
543. Goat
544. Goat
545. Goat
546. Car
547. Car
548. Goat
549. Goat
550. Car
551. Car
552. Goat
553. Goat
554. Car
555. Goat
556. Goat
557. Goat
558. Car
559. Goat
560. Car
561. Goat
562. Car
563. Car
564. Goat
565. Goat
566. Goat
567. Goat
568. Car
569. Goat
570. Goat
571. Goat
572. Goat
573. Goat
574. Car
575. Goat
576. Car
577. Goat
578. Goat
579. Goat
580. Goat
581. Goat
582. Car
583. Goat
584. Goat
585. Goat
586. Goat
587. Car
588. Car
589. Car
590. Goat
591. Goat
592. Car
593. Car
594. Car
595. Car
596. Goat
597. Car
598. Goat
599. Car
600. Goat
601. Goat
602. Goat
603. Goat
604. Goat
605. Goat
606. Goat
607. Car
608. Goat
609. Car
610. Car
611. Goat
612. Goat
613. Goat
614. Car
615. Goat
616. Goat
617. Goat
618. Goat
619. Goat
620. Car
621. Goat
622. Goat
623. Goat
624. Goat
625. Car
626. Goat
627. Car
628. Car
629. Goat
630. Goat
631. Car
632. Goat
633. Goat
634. Car
635. Goat
636. Car
637. Goat
638. Car
639. Goat
640. Goat
641. Goat
642. Goat
643. Goat
644. Goat
645. Goat
646. Goat
647. Goat
648. Goat
649. Goat
650. Car
651. Goat
652. Car
653. Goat
654. Goat
655. Goat
656. Car
657. Goat
658. Car
659. Car
660. Goat
661. Goat
662. Goat
663. Goat
664. Car
665. Goat
666. Car
667. Goat
668. Goat
669. Goat
670. Car
671. Goat
672. Car
673. Goat
674. Goat
675. Goat
676. Goat
677. Goat
678. Goat
679. Goat
680. Goat
681. Goat
682. Car
683. Goat
684. Car
685. Goat
686. Goat
687. Car
688. Car
689. Goat
690. Goat
691. Car
692. Goat
693. Car
694. Goat
695. Car
696. Goat
697. Car
698. Car
699. Car
700. Goat
701. Goat
702. Goat
703. Car
704. Goat
705. Goat
706. Car
707. Car
708. Goat
709. Goat
710. Car
711. Car
712. Car
713. Car
714. Goat
715. Goat
716. Car
717. Car
718. Goat
719. Car
720. Car
721. Goat
722. Car
723. Goat
724. Goat
725. Car
726. Goat
727. Car
728. Goat
729. Car
730. Goat
731. Goat
732. Goat
733. Goat
734. Goat
735. Goat
736. Goat
737. Car
738. Car
739. Car
740. Goat
741. Goat
742. Goat
743. Goat
744. Goat
745. Car
746. Goat
747. Goat
748. Goat
749. Goat
750. Goat
751. Car
752. Car
753. Goat
754. Goat
755. Goat
756. Goat
757. Goat
758. Goat
759. Goat
760. Goat
761. Goat
762. Goat
763. Goat
764. Goat
765. Car
766. Car
767. Goat
768. Goat
769. Goat
770. Goat
771. Goat
772. Car
773. Goat
774. Goat
775. Goat
776. Goat
777. Goat
778. Goat
779. Goat
780. Car
781. Goat
782. Car
783. Car
784. Car
785. Car
786. Goat
787. Goat
788. Goat
789. Goat
790. Goat
791. Goat
792. Goat
793. Car
794. Car
795. Car
796. Car
797. Goat
798. Goat
799. Goat
800. Goat
801. Goat
802. Goat
803. Car
804. Goat
805. Car
806. Goat
807. Car
808. Goat
809. Goat
810. Goat
811. Car
812. Goat
813. Goat
814. Goat
815. Car
816. Car
817. Goat
818. Car
819. Goat
820. Goat
821. Goat
822. Goat
823. Car
824. Car
825. Car
826. Goat
827. Goat
828. Car
829. Goat
830. Goat
831. Goat
832. Goat
833. Car
834. Goat
835. Car
836. Goat
837. Car
838. Goat
839. Goat
840. Goat
841. Goat
842. Car
843. Goat
844. Goat
845. Car
846. Goat
847. Goat
848. Goat
849. Goat
850. Goat
851. Car
852. Goat
853. Goat
854. Goat
855. Goat
856. Goat
857. Goat
858. Car
859. Goat
860. Goat
861. Goat
862. Goat
863. Goat
864. Car
865. Goat
866. Car
867. Goat
868. Goat
869. Goat
870. Car
871. Goat
872. Goat
873. Car
874. Goat
875. Goat
876. Goat
877. Goat
878. Car
879. Car
880. Goat
881. Car
882. Goat
883. Goat
884. Goat
885. Goat
886. Car
887. Car
888. Goat
889. Goat
890. Car
891. Goat
892. Goat
893. Car
894. Goat
895. Goat
896. Goat
897. Goat
898. Goat
899. Car
900. Car
901. Goat
902. Goat
903. Car
904. Goat
905. Car
906. Goat
907. Goat
908. Goat
909. Goat
910. Car
911. Car
912. Goat
913. Goat
914. Goat
915. Goat
916. Goat
917. Car
918. Goat
919. Car
920. Goat
921. Goat
922. Car
923. Car
924. Car
925. Goat
926. Goat
927. Car
928. Car
929. Car
930. Goat
931. Goat
932. Goat
933. Goat
934. Car
935. Car
936. Goat
937. Car
938. Goat
939. Goat
940. Goat
941. Goat
942. Goat
943. Goat
944. Goat
945. Goat
946. Goat
947. Car
948. Car
949. Car
950. Goat
951. Car
952. Goat
953. Car
954. Car
955. Goat
956. Car
957. Goat
958. Goat
959. Goat
960. Goat
961. Goat
962. Car
963. Car
964. Car
965. Car
966. Goat
967. Car
968. Car
969. Goat
970. Goat
971. Goat
972. Goat
973. Goat
974. Goat
975. Goat
976. Car
977. Goat
978. Goat
979. Goat
980. Goat
981. Car
982. Goat
983. Goat
984. Goat
985. Car
986. Goat
987. Car
988. Car
989. Car
990. Goat
991. Car
992. Car
993. Goat
994. Car
995. Car
996. Goat
997. Goat
998. Goat
999. Goat
1000. Goat

But what if there were 10 doors with nine goats and one car? The game would still work the same where the player picks one door, and the host reveals 8 goats. Should the player switch? Here is a 1,000 iteration simulation where there are 10 doors and the player always sticks with their original choice:

Player won 102 times and lost 898 times.

So clearly, the number of doors influences the probability that the player would be better of by switching to the other remaining door.

Would you like to run a simulation? Enter your variables below:

Custom test result

Player won 33 times and lost 67 times.

Methodology

This page uses the following PHP function to run the simulations:

function runTest($iterations, $number_of_doors=3, $player_stays=true, $show_pictures=false){

	// sanitize iterations
	$iterations = (int) preg_replace("/[^0-9]/","",$iterations);
	$iterations = ($iterations < 1)? 10 : $iterations;
	$iterations = ($iterations > 10000)? 10000 : $iterations;

	// sanitize number_of_doors
	$number_of_doors = (int) preg_replace("/[^0-9]/","",$number_of_doors);
	$number_of_doors = ($number_of_doors < 3)? 3 : $number_of_doors;
	$number_of_doors = ($number_of_doors > 10000)? 10000 : $number_of_doors;

	// generate a standard array
	$doors_original[0] = 'car';
	for($i=1; $i < $number_of_doors ; $i++){
		$doors_original[$i] = 'goat';
	}
	
	// initialize our scores
	$losses = 0; $wins = 0;
	
	// decide how big the images will be
	if($iterations >= 1000){
		$size = '24';
	} else if($iterations >= 100){
		$size = '32';
	} else {
		$size = '64';
	}
	
	$output = '<ol>';
	for($i=0; $i<$iterations; $i++){
		// make a temporary copy for this usage
		$doors = $doors_original;
		
		// shuffle the array
		shuffle($doors);
		
		// player always picks door number 1
		$players_first_choice = $doors[0];
		
		// did the player win?
		if($players_first_choice == 'car' && $player_stays==true){
			$output .= '<li class="c'.$size.'"><b>Car</b></li>';
			$wins++;
		} else if($players_first_choice == 'goat' && $player_stays==true){
			$output .= '<li class="g'.$size.'"><i>Goat</i></li>';
			$losses++;
		} else if($players_first_choice == 'car' && $player_stays==false){
			$output .= '<li class="g'.$size.'"><i>Goat</i></li>';
			$losses++;
		} else if($players_first_choice == 'goat' && $player_stays==false){
			$output .= '<li class="c'.$size.'"><b>Car</b></li>';
			$wins++;
		}
	}
	$output .= '</ol><div class="clear"> </div>';
	
	echo '<h3>Player won '.$wins.' time'.(($wins >1)? 's':'').
	     ' and lost '.$losses.' time'.(($losses >1)? 's':'').'.</h3>';
	if($show_pictures){
		echo $output;
	}
}