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 1 time and lost 9 times.

1. Goat
2. Car
3. Goat
4. Goat
5. Goat
6. Goat
7. Goat
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 344 times and lost 656 times.

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